Asset Measurement, Loanable Funds and the Cost

of Capital

EDWIGE CHEYNEL∗

BEATRICE MICHAELI

Very Preliminary and Incomplete

Columbia University

October 11, 2012

Abstract

In the capital market, accounting measurements of assets serve two primary func-

tions, to help the firm access loanable funds by pledging its own assets as collateral,

and to identify and liquidate unproductive assets whose return is below the cost of

capital demanded by investors. This paper shows that these two roles are generally

in conflict. The optimal measurement rule reports asymmetrically either low or high

assets to fulfill the dual role of facilitating credit and/ or to guide firms toward value-

enhancing investments. When credit is widely available, i.e., the cost of capital is

low, an accounting rule prescribing write-up measurements of high-value assets is

optimal. However, as credit tightens, the optimal measurement system will feature

impairments over low-value assets. After a collateral squeeze (e.g., a reduction in

the value of existing assets), the cost of capital increases with write-up accounting

and decreases with impairment accounting. After a credit crunch (e.g., a reduction

in available loanable funds), the cost of capital and the precision of the measurement

systemincreases.

∗Contact author: ec2694@columbia.edu. Tel: (212)-851-5863. Fax: (212)-316-9219. Columbia Busi-ness School, Columbia University, 3022 Broadway, New York, NY 10027.

1

1 Introduction

Firms are, more than ever, evolving in a global economy, where their investment choice

is contingent not only on the financing of their own project but also the attractiveness

of outside investment opportunities. The efficient capital allocation in this economy is

facilitated by mandatory disclosure, based on accounting measurement rules. As explic-

itly expressed in the objectives of the U.S. Securities and Exchange Commission,“...SEC

requires public companies to disclose meaningful financial and other information to the

public... The result of this information flow is a far more active, efficient, and trans-

parent capital market that facilitates the capital formation so important to our nation’s

economy.”1

This paper examines whether accounting measurement rules can improve investment

efficiency when firms do not observe the value of their assets that they can pledge to cap-

ital providers. If access to financing for economically viable productive investments is

undeniably one facet of the inefficiencies that regulation faces, evaluating the potential of

productive investments relative to outside investment opportunities is equivalently crucial

in selecting value enhancing investments. We study a simple general equilibrium model in

which accounting measurement rules condition firms’ investment choice between a pro-

ductive investment in a risky technology subject to moral hazard and external financing,

and a financial investment that returns the cost of capital. In particular, we derive the op-

timal information system that reports asymmetrically low and high assets and its complex

feedbacks with cost of capital, access to external financing and investment efficiency. We

analyze the dual role of the information system in alleviating inefficiencies - it facilitates

credit and guides firms in optimally choosing value-enhancing investments. We further

explore capital tightening shocks and their consequences on investment efficiency, cost of

capital and accounting measurement rules.

Goex and Wagenhofer (2009) has shown that mandatory disclosure reporting firms

with low assets is welfare improving when absent any information firms cannot run their

economically viable projects. Firms disclosing their low assets are squeezed out and allow

the remaining non-disclosing firms to meet the lenders’ collateral requirements for financ-

ing. The optimal regulation maximizes the amount of projects funded but cannot succeed

to finance all the firms due to moral hazard. We enrich this framework by introducing an

outside investment opportunity that offers the cost of capital similar to Holmstrom and

Tirole (1997). If a firm does not run the project, it invests in the open market. This com-

peting financial investment introduces an additional tension that might deter firmsfrom

1See http://www.sec.gov/about/whatwedo.shtml.

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running the project even if they can find financing and put forward the role of capital mar-

kets in determining the optimal information system and aggregate investment. Given that

the resource cost for a firm to run the project diminishes in the value of its assets, the pro-

ductive investment is less attractive for capital rich firms, and information on high assets

help capital rich firms to optimally select the efficient investment. Alternatively capital

rich firms have assets with high values but are unproductive. As a result, regulation is

confronted with inefficiencies that might require opposing measurements to be mitigated.

Once we model capital markets, we want to understand how measurement rules, cost

of capital and investment efficiency are intertwined. To this end, we first take the cost of

capital as given, then endogenously determine the supply and cost of capital.

The starting point of this paper consists of analyzing the optimal measurement rules

and the investment efficiency when the supply is perfectly elastic and firms’ investment

choice does not affect the cost of capital. The analysis reveals that the determining factors

in the selection of information systen are the level of aggregate wealth in the economy

and the financing conditions dependent on the cost of capital. When the economy is

wealthy and all firms can find financing without information, measurement of high values

is optimal and low values of costs of capital allow to achieve first best. For high values of

cost of capital, some capital rich firms do not measure their assets and run the productive

investment, whereas had they learned their assets’ value, they would have invested in the

financial opportunity. When aggregate wealth decreases, measurements of high assets

remains optimal only for low costs of capital, and are substituted by measurements of

low assets for high costs of capital as the aggregate wealth is insufficient to guarantee

financing. When the aggregate wealth further drops, no matter the cost of capital, only

measurements of low assets are optimal. Measurements of low assets partially offset the

productive distortion of capital poor firms at the expense of capital rich firms which take

the inefficient investment, the productive investment. We refer to measurements of low

(high) assets as impairment like (write-up) measurements. In all scenarios, we find that

optimal measurement rules prescribe partial information where non disclosing firms run

the productive investment and disclosing firms take the financial investment.

We next expand the analysis to a more realistic framework by considering the supply

and the cost of capital as endogenous. This general equilibrium approach takes into ac-

count that resources in the economy are limited. To this end, we assume that firms running

the productive investment are financed by firms foregoing the project and an exogenous

supply. Our results in general equilibrium can be diametrically different in partial equi-

librium. In particular write-up measurement thresholds decrease (more measurement) in

general equilibrium in response to a capital tightening shock while increase (less mea-

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surement) in partial equilibrium. The shrinkage in capital is adjusted through the cost of

capital that in turn affects the optimal measurement thresholds. Also unlike partial equi-

librium, write-up and impairment like measurement thresholds are unaffected by changes

in the project’s attributes in general equilibrium as the cost of capital fully offsets these

effects. The endogenous adjustment of the cost of capital to changes in macroeconomic

fundamentals is thus driving the results.

As a consequence the interaction between the endogenous cost of capital and ag-

gregate wealth is again a key determinant for the existence of a single stable general

equilibrium or two stable general equilibria. If the economy is wealthy and without in-

formation firms can find financing, the optimal cost of capital coincides with the cost

of capital without moral hazard, and implements the efficient investment by measuring

assets with high values. In contrast, if the economy is poor and absent any information

firms cannot find financing, the optimal cost of capital is associated with measurements of

low assets and cannot implement the efficient investment. There exists an economy that

is half way between the rich and the poor economy, that admits simultaneously the two

stable costs of capital. In this economy the cost of capital associated with measurements

of high assets and implementing first best, is lower than the cost of capital associated

with measurements of low assets. Both equilibrium costs of capitals are affected sim-

ilarly by changes in the project’s attributes. More attractive productive projects or less

moral hazard call for higher equilibrium cost of capital to counterbalance the increased

demand. However depending on the nature of the macroeconomic shock, their response

might differ. If we consider additional capital providers outside the financing firms and

their amount of capital suddenly decreases, there is a credit crunch. To counterbalance

this shock, both equilibrium costs of capital increase. Counter-intuitively, they respond in

opposite directions to a collateral squeeze: the cost of capital related to write-up measure-

ments increases while the equilibrium cost of capital related to impairment measurements

decreases. So a reduction of internal resources affects differently the costs of capital rela-

tively to a reduction of external resources. However these capital tightening shocks have

similar consequences on the aggregate investment by reducing the productive investment

and expanding measurements.

Our paper builds on the insights of three main strands of prior literature: cost of capi-

tal, disclosure and debt financing. Recent analytical and empirical research on accounting

rules, regulation and cost of capital focuses on the interaction between information pre-

cision in the economy and cost of capital. More specifically, some papers, e.g. Diamond

and Verrecchia (1991), Easley, Hvidkjaer, and O’Hara (2002), Easley and O’Hara (2004),

Lambert, Leuz, and Verrecchia (2007), Lambert, Leuz, and Verrecchia (2010), Chris-

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tensen, de la Rosa, and Feltham (2010) and Bertomeu, Beyer, and Dye (2011), study the

relation between information asymmetry among investors and cost of capital. In contrast

other studies, e.g. Botosan (1997), Core, Guay, and Verdi (2008), Cheynel (2013), model

homogeneous investors and look at their perception of disclosure on the cost of capital.

Our paper contributes to this literature by studying the impact of the optimal information

system on the cost of capital when firms and investors share the same information and

there is asymmetry of disclosures across firms leading to different investment decisions.

The link between the type of accounting disclosure and debt financing has been stud-

ied extensively. For example, Beyer (2012) considers aggregate reports of asset values

and shows that debt contracts are more efficient when covenants are written in terms

of conservative reports as long as the required capital is not too high. Otherwise, fair

value accounting may lead to more efficient debt contracts. Other papers that consider

the preferred accounting system in a single firm economy are, for example, Smith (2007),

Gigler, Kanodia, Sapra, and Venugoplan (2009), Li (2012), Caskey and Hughes (2012).

Our paper endogenizes the accounting information system and finds the optimal one in a

multiple firm economy. In that respect our paper is closely related to Goex and Wagen-

hofer (2009) who show that absent outside investment opportunity and capital markets it

is optimal to set mandatory disclosures on low value assets that will allow firms with high

value assets to fund their projects. Our paper prescribes the optimal information system in

conjunction with fundamentals of the economy and firms’ investment opportunities and

offers new implications to the current findings.

The idea that disclosure affects not only market prices but also production and invest-

ment decisions is central to the real effects of accounting literature. It was first pioneered

by Kanodia (1980) and further analyzed in Kanodia and Lee (1998), Kanodia, Sapra, and

Venugopalan (2004), Melumad, Weyns, and Ziv (1999), Kanodia, Mukherji, Sapra, and

Venugopalan (2000), Sapra (2002), Plantin, Sapra, and Shin (2008). Gao (2010) further

studies the link between cost of capital and real effects of information precision. While

Gao (2010) focuses on risk sharing issues due to measurements in a single firm economy,

our study stresses the importance of limited resources in a multiple-firm economy and the

effect on the cost of capital and investment efficiency.

The notion of limited resources is a core ingredient in general equilibrium literature.

We endogenously determine the supply and cost of capital since resources are limited, as

part of the firms become financing firms and transfer their assets to the entrepreneurial

firms. In that sense our paper is related to Boot and Thakor (1997) who create a resource

transfer theory in a model with postlending moral hazard that can affect payoffs to cred-

itors. While Boot and Thakor (1997) concentrate on determining which borrowers will

5

turn to banks and which to capital markets for financing of their project, we endogenously

derive investigate which firms become financing firms and entrepreneurial firms respec-

tively. In our model the firm’s ability and preference to become provider or user of capital

is determined based on an endogenous information system and a resource constraint.

Limited resources and credit constraints create the need for intermediaries. Papers

investigating the role of intermediaries as efficient providers of liquidity and/or monitors

in a general equilibrium framework are, for example, Chen (2001), Michelacci and Suarez

(2004), Allen and Gale (2004) etc. The paper most closely related to ours is Holmstrom

and Tirole (1997). The focus of their paper is to analyze how, in a general equilibrium

setting, the distribution of wealth across firms, intermediaries, and uninformed investors

affects investment, interest rates, and the intensity of monitoring when the value of the

assets are common knowledge. We assume instead that firms do not observe the value

of their assets and focus on the optimal measurement rules and their interaction with

access to financing, investment opportunities and the cost of capital in the context of

capital markets. By doing so we expand the general equilibrium framework to accounting

measurements.

The paper proceeds as follows. Section 2 outlines the economic setting, Section 3

describes the optimal information system with exogenous cost of capital and Section 4

solves for the equilibrium cost of capital.

2 Model

There is a continuum of firms in the economy. These firms can either invest in a project

(thereafter: “productive investment”) and become entrepreneurial firms or invest in the

open market (thereafter: “financial investment”) and become financing firms.2 Firms are

run by risk neutral owners-managers who are protected by limited liability. All firms have

access to the same technology but they are endowed with a different amount of unob-

servable pledgeable assetsA. The distribution of assets across firms follows a probability

density functionf(.) > 0 and a cumulative density functionF (.), which are common

knowledge. To run the project entrepreneurial firms need to raise debt and pledge their

assets to guarantee financing. Before entering the debt contract, firms establish an infor-

mation system that reports information about their pledgeable assets. Firms updatetheir

2We assume that firms cannot invest simultaneously in their project and provide capital. If the firm couldoperate the new project by selling its assets, it would mean that the old and new projects of the firm wouldbe independent of each other. This discussion is at the essence of the theory of the firm. We view the firm asan economic entity that needs to manage old and new projects simultaneously. When it comes to financingthe new project, this decision is made at the firm level, coordinating the old and new projects.

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belief regarding the value of their assets conditional on the information released from the

information system and choose the productive investment or the financial investment.

The sequence of the events is as follows. At date0 the firms are endowed with unob-

servable assets. At date1 an information system is set and the assets are valued either at

A (for disclosing firms) or at the conditional expectation given no disclosureE(A|ND)

(for non-disclosing firms) . At date2 investment choices are made and contracts between

entrepreneurial firms and lenders are signed. At date3 the outcomes are realized.

Figure 1: Timeline

Assume that the entrepreneurial firm needsI > 0 of capital to finance a new project,

which is obtained from a competitive market with risk-neutral lenders. The firm has some

assetsA ≥ 0 independent from the new project and which can be pledged as collateral.

The assets cannot be liquidated to finance the project. The firm needs these assets to

maintain the good functioning of its old and new projects. Assume thatA is not observable

absent any information system. We assume that these pledgeable assets are either cash or

cash equivalents, or assets that can be liquidated. The uncertainty of the measurement of

these assets is a common issue in practice.3 The project can yield two outcomes “Success”

with valueH > 0 and “Failure” with value normalized to zero. The project succeeds with

probabilityp ∈ (0, 1) if the manager provides efforte = 1 and succeeds with probability

p−Δp, whereΔp ∈ (0, p) if the manager provides no efforte = 0. The cost of effortis

3A precise assessment of the assets requires a careful analysis and inventory of all the pledgeable assets.Existing transactions on similar assets might be hard to find. Even a correct assessment of cash inside alarge group is an issue. Tracking all the cash transfers between the different entities of the firm requirescareful checking of these flows by reconciling the different accounting systems.

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c > 0. The project is only economically viable if the manager exerts effort, i.e.pH − c−

γI > 0 > (p−Δp)H−γI, whereγ is the expected rate of return required by the market.

There is no heterogeneity in the projects across firms.4 A capital rich firm has the same

project as a capital poor firm.5

The entrepreneurial firm issues debt with face valueF ∈ [I,H ]. Conditional on

the high outcome, it pays backF and conditional on the low outcome, it transfers all

pledgeable assets to the lenders. In case of a success the firm keeps its assets and values

them atηA, whereη ≥ 1 reflects the appreciation of its assets if they are not liquidated.

The parameterη captures the difference between the retention value and the liquidation

value of the pledgeable assets. There are two main reasons for this difference. On one

hand, lenders face information asymmetry and incur transaction costs to liquidate the

firms’ assets. On the other hand, firms create synergies between old and new projects.

This interpretation is in line with Goex and Wagenhofer (2009). Firms’ ability to enhance

the value of their assets is increasing the minimum expected rate of return on the financial

investment, that deters some of the firms to run the project. Firms that do not run projects,

invest in the open market and demand a rate of returnγ > γmin ≡ ηp + (1 − p).6 We

consider values of cost of capital such that the financial investment is at least attractive

for some firms if they know the value of their pledgeable assets. The financial investment

offers a safe alternative to the firm and a saving on effort cost. At the same time if the

firm forgoes the project, it also gives up on the upside potential to receive cash flows, if

the project succeeds. The resource cost for a firm to become an entrepreneurial firm isA,

and thus the opportunity cost of becoming an entrepreneurial firm equal toγA increases

with firms’ initial wealth. Equivalently, firms with less wealth have a relatively more

productive entrepreneurial technology.

Regulation might improve the functioning of capital markets by helping firms and

lenders acquire information necessary to guarantee trading and common agreements. To

abide by the rules, firms can commit to implement an accounting system that provides

information about the firm’s assets before the negotiation of the debt contract.7 The in-

4Therefore project outcomes are assumed to be perfectly correlated.5The value of the collateral is not correlated with the type of project. There is little evidence to support a

positive relation between good projects and capital rich firms. Capital rich firms own usually a lot of assetsas they are more mature and less innovative on average, and have fewer new projects to run. If capital richfirms have worse projects, this would reinforce the findings of our paper.

6At γ ≤ γmin, all the firms in the economy prefer the productive investment.7Alternatively we can assume that the lenders implement an accounting system to prevent firms from

lying about the outcome of the information system. An auditor can also be hired to set up the mandatoryinformation system. Each one of these assumptions implies that the firms do not have superior knowledgeabout the value of their assets than the investors.

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formation system is defined by a functionθ(A) ∈ [0, 1] such thatA is measured with

probability 1 − θ(A) and not measured with probabilityθ(A). The firm can credibly

commit to measure and disclose the information and cannot bias the information.8 The

information on the pledgeable assets is common knowledge. Post measurement, firms

differ in the measurement of their pledgeable assets. If the amount of pledgeable assets

is not measured, the amount of pledgeable assets is valued atE(A|ND), the conditional

expectation after no disclosure. If the amount of pledgeable assets is measured, the value

of the assets isA whereA ∈ R+.9 After observing the measurement outcomes, firms

choose between the productive and the financial investment.

Finally the choice between the productive and the financial investment determines the

cost of capital and is at the essence of capital markets’ functioning. As a first path, we

assume thatγ is exogenous and that there is an infinite supply of financial investment

opportunities that returnγ. We later determine endogenously the level of cost of capital

that clears the markets at date2.10 Restraining the supply might represent a more autarchic

economy in comparison to a perfectly elastic supply in a small open economy. To relate

information system, investment efficiency and cost of capital to macroeconomic shocks,

we model two types of macroeconomic shocks tightening capital that occur between date

0 and date1. First we model a collateral squeeze as a proportional decrease in the assets

of all firms in the economy by a shareε ∈ (0, 1). The collateral squeeze also reduces the

aggregate wealth in the economy byε. When we consider the endogenous cost of capital,

we further introduce an additional source of capital tightening. The supply comes from

financing firms abandoning the project and also from an exogenous supplyK ∈ [0, I ].11

We interpret a reduction in this exogenous supply byκ ∈ (0, 1) as a creditcrunch.

8If the manager of the firm lies and investors or regulatory institutions discover that he has inappropri-ately distorted the information, he would be severely punished. Furthermore, a deliberate failure to measureinformation is more reprehensible under mandatory disclosure than under voluntary disclosure.

9We assume that disclosing any event is cost free. We isolate the driving forces relating measurementand the cost of capital from any effect due to cost specifications. Adding a fixed cost when firms measuretheir assets would not change qualitatively the results. However if the cost is a function of the value of theassets, the optimal information system would change and not surprisingly, low cost events would be morelikely to be measured.

10This happens at date2, event C in the timeline of Figure1.11If K > I , there would be excess supply and an equilibrium with a negative interest rate would exist.

More realistically these outside capital providers would invest in the open market so that the cost of capitalequals theγmin and then would keep their capital (hoarding their money rather than investing at a negativeinterest rate).

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3 Exogenous Cost of Capital

We first study the optimal information system if there is an infinite supply of financial

investment opportunities that offer a rate of returnγ. This assumption is standard in the

literature when the cost of capital is taken as given and the focus is on a partial equi-

librium.12 This analysis examines the impact of measurement and cost of capital in a

“partial equilibrium” framework, that is, within the context of a single market, neglecting

any induced effects on other markets due to a change in investment choice. Although

relevant only when such effects can reasonably be assumed to be unimportant, this ap-

proach shows how asset measurement rules depend on market conditions and how they

induce firms to prefer productive over financial investments and vice versa. This setting

also corresponds to a small open economy as discussed in the macroeconomic literature

where firms can borrow and lend without altering the world cost of capital.13 We study as

a benchmark for this partial equilibrium setting, the first best that determines the efficient

investment. We refer to this first best asunconstrained first bestthat one should observe if

a benevolent social planner has full information. We also derive the optimal information

system without moral hazard to single out the effects due to moral hazard in the rest of

the analysis. We then move to a second best analysis, taking the information system as

given. In section 3.3, we complete the partial equilibrium analysis by finding the optimal

information system with moral hazard.

3.1 Benchmarks

Unconstrained First-Best: Full Information

We assume that there is a benevolent social planner observing the level of effort and the

value of the pledgeable assets. He acts as a representative agent that optimally allocates

the type of investment across firms. The productive investment is selected when in expec-

tation it offers a better opportunity than the financial investment :pH − γI + (ηp + (1−

p))A − c > γA. The unconstrained first-best solution maximizes the sum of individual

firms’ utilities. Proposition 1 characterizes the efficient investment:

12See for example Modigliani and Miller (1958), Gennotte and Pyle (1991) and Mailath and Mester(1994), Arya and Glover (2003), Goex and Wagenhofer (2009) and Li, Liang, and Wen (2011).

13This assumption is also similar to a two period model, where the firms in the economy maximize theirutility by optimally shifting their consumption from one period to the next and create capital at a costγ bysubstituting their consumption in the first period. To see this let us define the firm’s utility byU = c1+ 1

γ c2.If the firm gives up one unit of consumption in the first period it will get1/γ in the future. The cost of capitalcan then also be a measure of the firms’ impatience.

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Proposition 1

(i) The first-best aggregate utility is

UFB =

∫ AH

0

(pH − γI + (ηp + (1− p))A− c)f(A)dA +

∫ ∞

AH

γAf(A)dA,

whereAH =pH − γI − c

γ − (ηp + (1− p))> 0 (2)

(ii) AH is increasing inp, H andη, and decreasing inγ, I andc.

(iii) In case of a collateral squeezeAH increases.

In equilibrium, there will be a cut-offAH that determines the type of investment the

firms make. Firms withA ≤ AH choose to run the project and become entrepreneurial

firms and those withA > AH choose to be financing firms. This result is related to the

assumption that projects are the same across firms and that if a firm chooses the financial

investment, it liquidates its assets14 and invest their money at the cost of capital. The direct

cash flow a firm receives by running the productive investment is independent of the size

of its initial wealth, i.e.pH − γI − c. In contrast, a firm liquidating its assets invest the

integrality of its wealth in the outside investment opportunity that offers the same return

per unit of investment to all the firms. The magnitude of the outside gains is equal to

γA and will differ across firms should they choose the outside investment. Hence, a firm

with a high (low) value of assetA has a comparative advantage in becoming a financing

(an entrepreneurial) firm. This unconstrained first best generalizes the first best in Goex

and Wagenhofer (2009). In their paper the cost of capital is normalized to1 and thus the

social surplus is maximized if all firms run the productive investment.

The heterogeneity in wealth might capture the firms’ maturity. In practice firms start-

ing their activities do not own a lot of assets and need external financing to grow their

business. Mature firms are more likely to have accumulated assets and if they choose

to run a new project, it needs to add value to the entire business. Even if the project is

economically viable, the firm needs to retain its assets to run it and forgoes the return

on the outside investment opportunity. The model predicts that firms that liquidate their

assets, have accumulated a lot of wealth and their new project does not create enough syn-

ergies with the older projects to justify further productive investment in the same business

14Broadly speaking the liquidation of these assets does not necessarily mean the termination of the firmper se but the termination of its old business model.

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relatively to the outside investment opportunity.15

More firms with valuable assets lead to more aggregate financial investment. There-

fore, when the value of the assets is reduced by the same proportion for all firms and the

aggregate wealth in the economy decreases it is immediate to observe that the efficient

investment choice shifts toward more productive investment. If this reduction in asset

value is interpreted as a collateral squeeze and the world cost of capital is not affected, the

productive investment is increasing. This counterintuitive effect relies on the assumption

that the cost of capital is maintained at the same level because more supply of outside in-

vestment compensates for the squeeze in collateral. The remaining comparative statics are

intuitive: if the project is more risky (lowerp andH or higherI) or more costly (higher

c) or the financial investment is more attractive (higherγ) or the firm creates less synergy

(lowerη), the cut-offAH decreases and so does the aggregate productive investment.

Optimal Information System without Moral Hazard

We now derive the optimal information system when there is no moral hazard but as-

sets’ values are not observable. This second benchmark removes the distortions due to

moral hazard. Absent any information on assets’ values the first best is not attained as

firms are pooled and take the same investment. Information can resolve the investment

inefficiencies by screening out the capital rich firms as the disclosing firms (A > AH).

The remaining non disclosing firms will then invest in the productive investment. Any

information system that measures the assets’ values aboveAH will attain first best. The

range of these information systems vary from full information to a write-up measurement

of assets aboveAH . If there is a small cost to measure information disclosing assets only

aboveAH would be optimal.

Proposition 2 If there is no moral hazard but the assets are not observable, any informa-

tion system prescribing measurement of assets aboveAH is optimal and the first best is

attained.

Without moral hazard firms will exert effort and find financing should they invest.

Knowing the value of their assets, firms at the top will select the financial investment and

non disclosing firms will in turn select the efficient investment, the productive investment.

15Our model does not refer to firms in financial difficulty, liquidating their assets to meet their obligationsor because their old projects were most likely not profitable. In our setting firms liquidate their assetsbecause they want to realize the gains of past profitable projects and the future project does not bring enoughreturns overall. In other words they keep their assets if they keep growing as these assets are necessary fortheir business model.

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Firms at the top need to know the liquidation value of their assets to invest in the financial

investment. write-up measurements are thus not in contradiction with the conservative

principle as disclosing firms liquidate their assets to realize their gains. They report gains

only when there is no uncertainty remaining on the value of their assets.

3.2 Moral Hazard, Full Information and No Information

Moral Hazard and Full Information

In the second best, lenders do not observe effort should the firm run the project. We first

consider a case in which the information system releases full information. The lenders

will only finance the project if they are protected against downside risk, while the firm

will run the project if and only if the project is funded and returns more value than the

financial investment. We first determine the maximum amount of pledgeable assets to

prefer the productive investment. Next we look at the minimum level of pledgeable assets

that a firm needs in order to obtain financing.

Productive Investment

Using a cost and benefit analysis we find the condition guaranteeing that a firm prefers

to run the project. The productive investment must return sufficiently high returns to deter

the entrepreneurial firm from investing in the open market (financial investment). The

entrepreneurial firm earns an expected utilitypH − γI + (ηp + (1 − p))A− c when un-

dertaking the project. The project remains attractive as long as the firm expects a higher

utility than investing in the open market:pH − γI + (ηp + (1 − p))A − c ≥ γA. This

condition is identical to the first-best condition and thus firms withA ≤ AH would opti-

mally run the project whereas firms withA > AH would select the financial investment.

From proposition 1, the financial investment is more attractive if it offers higher returns

or if the operating project provides lower cash flows, requires higher capital or the cost of

effort is large.

We next determine the condition guaranteeing the financing of the project. If the

entrepreneur-manager exerts high effort his utility is given by:

U = p(H − F + ηA)− c

The debt contract must discipline the manager of the firm into exerting high effort. To

ensure that the manager provides high effort, the incentive compatibility (IC) constraint

13

needs to be satisfied:

Δp(H − F + ηA)− c ≥ 0

For lenders to be willing to provide financing, they need to break even should the en-

trepreneurial firm invest.

pF + (1− p)A = γI

Thus, the face value of the debt must be equal to:F = γI−(1−p)Ap

. SubstitutingF in the

(IC) leads to:

A ≥ AL =1

ηp + (1− p)

(pc

Δp+ γI − pH

)

The debt contract secures the repayment of the loan with collateral. The firm will find

financing if its pledgeable assets are greater thanAL.16 We assume thatAL > 0 so that

firms need collateral to fund the project. This condition is identical to Holmstrom and

Tirole (1997) and corresponds topH − γI < pcΔp

, which means that the surplus of the

project is insufficient to compensate for the disutility to induce the manager to exert high

effort. The firm must be able to commit a significant portion of its own wealth to the

contract to convince the lenders to provide capital. This minimum amount of collateral

disciplines the firm to provide effort and run an economically viable productive invest-

ment and simultaneously protects lenders against downside risk. However, since firms

with pledgeable assets belowAL do not receive financing and cannot run the project, the

moral hazard creates a distortion for capital poor firms that are not financed.

Lemma 1 The minimum collateral to find financingAL is increasing inc, γ andI, and

decreasing inΔp, p, η andH. In case of a collateral squeezeAL increases.

Lenders require more collateral if the project is more risky (lowerp andH or higher

I), the moral hazard problem is more severe (higherc, smallerΔp), the financial invest-

ment is more attractive (higherγ) or the firm creates less synergies between its old and

new projects (lowerη). If there is a collateral squeeze, lenders require higher amount of

pledgeable assets and capital poor firms will thus be more affected by a collateral squeeze,

as they will find it harder to achieve funding. Combining the maximum asset requirement

to prefer the productive investment and the minimum collateral to find financing we derive

the condition for the productive investment to be undertaken:

Proposition 3 There exists a uniqueγmax such thatAH(γmax) = AL(γmax). γmax is

decreasing inc andI and increasing inH, η, Δp andp, and is not affected by a collateral

squeeze.

16As firms are risk neutral, they are indifferent to pledgeAL or all their assets, should they invest.

14

(i) If γ ≤ γmax, firms withAL ≤ A ≤ AH run the productive investment. The remain-

ing firms choose the financial investment. The amount of projects in the economy is

increasing inH, Δp, p andη, and decreasing inc, γ and I. A collateral squeeze

has ambiguous effects on the aggregate amount of projects.

(ii) If γ > γmax, none of the firms runs the productive investment. All firms invest in the

open market.

The minimum of pledgable assets required by lenders,AL, can be greater than the

maximum of pledgeable assetsAH that the firm is willing to give as collateral to invest in

the project. However, in this case none of the firms undertakes the productive investment.

The investment choice will differ across firms if and only ifAL < AH . Firms withA <

AL prefer the productive investment but short of financing take the financial investment.

Firms with A > AH prefer the financial investment. Thus firms withAL < A < AH

run the operating project. The aggregate productive investment is larger when financing

requirements are looser and the outside investment opportunity is less attractive. These

two conditions are simultaneously met ifAL moves downward whileAH moves upward

or does not change. Except for a collateral squeeze, these two cut-offs are inversely

affected by exogenous parameters. If the project is less risky (higherp andH or lowerI),

the moral hazard problem is less severe (lowerc), the financial investment is less attractive

(lower γ) or the firm creates more synergies between its old and new projects (higherη)

lenders require less collateral and at the same time firms need more collateral to prefer

the financial investment. Furthermore, when the probability of success for the project

changes a lot due to the exerted effort (highΔp), managers have less incentives to shirk

and hence lenders require less collateral (lowAL). However, the change in probability of

success does not affect the attractiveness of the outside investment opportunity (no change

in AH), because once the firm runs the project, it will provide effort in equilibrium and its

investment choice does not depend onΔp . Thus, the higher the incremental contribution

of the exerted effort, the higher the aggregate productive investment.

In contrast, the effect of a collateral squeeze on the aggregate productive investment

is ambiguous. It increases simultaneously the minimum collateral requirement and the

maximum collateral to prefer productive investment. As a result, both capital rich firms

and capital poor firms are hit. Firms with initial assetsA ∈ [AL, AL

1−ε] cannot find financ-

ing anymore. This effect of the collateral squeeze replicates the effect in Holmstrom and

Tirole (1997) . The new prediction relates to the effect on capital rich firms: the firms

with initial collateralA ∈ [AH , AH

1−ε] switch to the productive investment. If the propor-

tion of capital rich firms preferring the productive investment outweighs the proportion

15

of capital poor firms squeezed out, the amount of productive investment increases. Left-

skewed distributions are more likely to result in more productive investment. If the ratio

of entrepreneurial firms over financing firms declines, a collateral squeeze pushes firms

to seek the financial investment.

Distortions with Full Information

It follows from Proposition 3 that full information is not efficient. On one hand,

if AL > AH all the firms withA < AH take the financial investment whereas they

should have taken the productive investment. On the other hand, ifAL ≤ AH firms with

A < AL cannot find financing for their project and are hence forced to take the financial

investment. This distortion at the bottom is worsened by a collateral squeeze. Everything

else equal, ifAL < AH , a collateral squeeze affecting all the firms will prevent more

capital poor firms from producing asAL increases. It will also attract more capital rich

firms to run the project but this effect coincides with the first-best investment choice. As a

result, a reduction in asset value has adverse effects on the amount of aggregate productive

investment but clearly worsens the distortion at the bottom. IfAL > AH , the distortion

on the productive investment is unaffected by a collateral squeeze.

We want to investigate whether we can find an information system that can alleviate

the investment distortion. On one hand, ifAL < AH (i.e if γ < γmax) an information

system that does not report information for firms with asset valueA ∈ [AL − δ, AH ]

whereδ is small, will improve the productive investment. Non disclosing firms can find

financing and run the project. Firms with asset valueA ∈ [AL − δ, AL] pooled with

capital richer firms take the efficient investment. On the other hand, ifAL > AH (i.e if

γ > γmax) providing only partial information cannot alleviate the distortion and all firms

take the financial investment. Lemma 2 summarizes these findings:

Lemma 2

(i) Full information creates underproduction distortion for firms withA < AL. A

collateral squeeze weakly worsens the investment distortion.

(ii) There exists an information system with partial measurement that weakly dominates

full measurement.

The observability of the assets cannot solve the distortion due to moral hazard. Pool-

ing firms with intermediate valuesAL ≤ A ≤ AH with some capital poorer firms mit-

16

igates the distortion at the bottom. We next investigate if the investment choice can be

optimal if there is no information about the pledgeable assets.

Moral Hazard and No Information

We keep unobservable effort and study the information system that provides no informa-

tion about the assets. We will later use this analysis as a helpful benchmark to evaluate

the incremental benefit of the optimal information system.

Productive Investment

If we do not observe the value of the assets, firms will not differ in their investment

choice. We can face two scenarios. IfAL < E(A) < AH the firms run the projects,

otherwise they take the financial investment.

Proposition 4 There exist uniqueγL andγH , given by:

γL ≡pH − pc

Δp+ E(A)(ηp + (1− p))

I

γH ≡pH − c + E(A)(ηp + (1− p))

E(A) + I,

so that:

(i) If γ ≤ min(γL, γmax, γH), all projects are financed and run by the firms. (AL <

E(A) < AH).

(ii) Else the firms conduct the financial investment.

(iii) Both γL and γH are increasing inH, η and p and decreasing inI and c. γL is

decreasing in a collateral squeeze and decreasing inΔp, whileγH is increasing in

a collateral squeeze.

The intuition behind this result is shown in Figure 2. Case 1 corresponds to a limiting

case where for all costs of capital the aggregate wealth defined asWmax is larger than

AH , and all firms prefer the financial investment. Case 3 is the cut-off scenario where the

aggregate wealth is equal toW imp = AL(γmax), that is, for all costs of capital firms find

financing and always prefer the productive investment. We end up with case 5 where the

aggregate wealth is equal to or less thanW low = AL(γmin) and for all costs of capital

firms short of financing take the financial investment. Case 2 and case 4 are intermediary

17

cases between the pivotal cases. From case 1 to case 3, productive investment continues

to expand up to a turning point when the wealth in the economy is equal to the minimum

collateral requirement for the highest cost of capital, defined asW imp = AL(γmax). At

W imp, no matter the level of cost of capital firms run the productive investment. From

any wealth belowW imp , productive investment shrinks until it completely disappears for

very low wealth in the economy short of financing (Case 5).

Distortions with No Information

Lack of information creates distortions in investment choice. All firms in the economy

are pooled: capital poor firms as well as capital rich firms run the project ifAL < E(A) <

AH (i.e. γ ≤ min(γL, γmax, γH)). The investment choice is suboptimal as capital rich

firms withA > AH should have taken the financial investment. There is overproduction in

the economy. Otherwise, all firms in the economy take the financial investment and firms

with A < AH would have been better off running the project, but absent any information

they cannot discriminate between investments. In this case there is underproduction in

the economy. These distortions depend on the cost of capital and the aggregate wealth.

Lemma 3

(i) No information creates investment distortion.

– If γ ≤ min(γL, γmax, γH) there is overproduction for all firms relative to first-

best.

– Otherwise there is underproduction for all firms relative to first-best.

(ii) DefineW shock as the aggregate wealth after the collateral squeeze,

– If W shock < W imp andγL(W shock) ≤ γ ≤ min(γL(E(A)), γH(E(A)), γmax),

firms switch from the productive investment to the financial investment. Else

firms keep the productive investment.

– If W imp > W shock > W low and the cost of capitalγH(E(A)) ≤ γ ≤

min(γH(W shock), γL(E(A)), γL(W shock), γmax) firms switch from the finan-

cial to the productive investment. Else firms keep the financial investment.

(iii) Partial information weakly dominates no information.

To understand the dynamics at play with a collateral squeeze, we change the aggregate

wealth without altering the pre-squeeze cost of capital. On one hand if we start with

18

overproduction (for example case 3), a collateral squeeze make the firms change to the

financial investment if the wealth stay aboveW low but falls belowW imp and the pre-

squeeze cost of capital is sufficiently large. Alternatively once the wealth drops below

W low, firms always switch to the financial investment. On the other hand if we start with

underproduction (for example Case 1), firms switch to the productive investment if the

wealth after the collateral squeeze is aboveW imp and the pre-squeeze cost of capital is

relatively low. AtE(A) > W shock = W imp, they all switch to the productive investment.

In all scenarios we can find an information system that achieves a better investment al-

location. IfE(A) < AL < AH setting up an information system that measures pledgeable

assets with valuesA ∈ [AL, AH ] is more efficient than no information. In this case the

disclosing firms will run the productive investment and the non disclosing firms will con-

duct the financial investment. This information system alleviates partially the distortion

on the productive investment for firms with intermediate pledgeable assetsA ∈ [AL, AH ].

On the other hand, ifAL < E(A) < AH and there is no information all firms run the

productive investment. An information system disclosing information for the firm with

the highest value of pledgeable asset dominates no measurement as the most capital rich

firm knowing its information will prefer the financial investment, which is the efficient

investment. The remaining non disclosing firms keep the productive investment.

3.3 Moral Hazard and Optimal Information System

We have established in the preceding section that forγ < γmax partial information is wel-

fare improving and dominates both full measurement and no measurement. This happens

because partial information might facilitate financing of projects or determine the type of

investment undertaken by the firm. In the remainder of the paper we restrict our attention

only to cost of capital values belowγmax, which implies thatAL(γ) < AH(γ).

Impairment-like Measurements vs Write-up Measurement

The optimal information system prescribes partial measurement and separates firms into

disclosing and non disclosing firms. Based on the value of their assets, the disclosing

firms will select the most value enhancing investment. The regulator needs to determine

the investment choice for the non disclosing firms - financial or productive investment.

Assuming that non disclosing firms take the financial investment, the best way to design

the information system would return the same inefficient allocation of investment across

firms as the one under full information. Hence, it is optimal to prescribe the productive

investment to non disclosing firms.

19

Lemma 4 The optimal information system prescribes the non disclosing firms to take the

productive investment.

Once the regulator knows that the optimal strategy is to ask non disclosing firms to

take the productive investment, he needs to determine the set of non disclosing firms.

Firms withA ∈ [AL, AH ] will alwaysbe non disclosing firms. The optimal information

system coincides with the first-best choice for those firms. The regulator is left with

determining the investment choice for the firms at the bottom (A < AL) and at the top

(A > AH). Firms at the bottom need to be non disclosing firms to find financing for their

project. In contrast firms at the top need to measure to take the efficient investment. The

regulator cannot usually implement the desired measurements for firms at the bottom and

at the top simultaneously. The issue remains whether it is optimal to offset the distortion

only for firms at the bottom and not improve the investment choice for firms at the top, or

vice versa. To this end, the regulator solves the following optimization program:

maxθ(A)

U(θ(A)) =

∫ AL

0

γA(1− θ(A))f(A)dA +

∫ ∞

AH

γA(1− θ(A))f(A)dA

+

∫ AH

AL

(pH − γI + (ηp + (1− p))A− c)(1− θ(A))f(A)dA

+

∫ ∞

0

(pH − γI + (ηp + (1− p))A− c)θ(A)f(A)dA

s.t.

∫∞0

Aθ(A)f(A)dA∫∞

0θ(A)f(A)dA

≥ AL if θ(A) 6= 0 for some A (3)

Constraint (3) allows financing for non disclosing firms so that they can run the

project. The optimal measurement rule depends on the wealth in the economy and the

investment distortion at the bottom is always mitigated. This feature of the optimal infor-

mation system stems from satisfying the constraint that secures financing for non disclos-

ing firms.

Proposition 5

(i) If E(A) < AL(γmax) = W imp, an impairment-like information system with impair-

ment thresholdAimp < AL is implemented.Aimp is uniquely defined by:

∫ ∞

Aimp

Af(A)dA− AL

∫ ∞

Aimp

f(A)dA = 0

Firms with A ≤ Aimp measure and do not undertake the project whereas firms

aboveAimp do not measure and undertake the project.Aimp is increasing inγ, I

20

andc and decreasing inH, η, p andΔp. In case of collateral squeeze the impair-

ment threshold increases.

(ii) If E(A) > AL(γmin) = W low, a write-up information system with write-up thresh-

old max{Aup, AH} is implemented.Aup is uniquely defined by:

∫ Aup

0

Af(A)dA− AL

∫ Aup

0

f(A)dA = 0

Firms withA ≥ max{Aup, AH}measure and do not undertake the project whereas

firms belowmax{Aup, AH} do not measure and undertake the project.Aup is in-

creasing inγ, I and c and decreasing inH, η, p and Δp. In case of collateral

squeeze the write-up threshold increases.

Proposition 5 describes two types of measurements - impairment-like measurement

and write-up measurement. Impairment-like measurements are more informative about

low type events, whereas write-up measurements are more informative about high type

events. Write-up measurements are less common in practice than impairment measure-

ments. A classic example is the capitalization of Research and Development expenses

once technological feasibility is met. This can be interpreted as a measurement of good

news. Furthermore, revenues (recognition of good news) are recognized when earned and

if it is reasonably expected that payment will be received no bad debt expense is rec-

ognized (no bad news). An additional example is the upward revaluation allowed under

IFRS17. If the economy is overall wealthy and the asset values are not expected to decline

this will lead to measurement of good events. A common feature of these three examples

is that the economic prospects are good (feasibility is met and the technology product is

likely to succeed, the client is expected to have sufficiently high wealth and be able to

pay, the asset is in good condition and the economy is strong so no reduction in value

is expected). If there is a small collateral squeeze (local change), the set of disclosing

firms decrease for write-up measurements whereas this set expands for impairment like

measurements. Intuitively non disclosing firms need to find financing and in response to

collateral squeeze, their conditional expectation after the release of information need to

increase to meet the tighter financing requirements. The measurement thresholds increase

to adapt to the shock. In contrast, the other comparative statics forAH andAup (or Aimp)

17IAS 16 Property, Plant and Equipmentallows firms to choose revaluation method to measure theirfixed assets. If a firm adopts the revaluation method and the value of an asset increases the firm can adjustupward the value of the asset. In the U.S. upward revaluation of fixed assets is not allowed as per FASBStatement No. 144,Accounting for the Impairment or Disposal of Long-Lived Assets

21

are diametrically different. If the project is more attractive (H, p, η higher orI or γ lower)

or the moral hazard is less severe (c or 1−Δp lower),Aup (or Aimp) decreases whileAH

increases. However these opposite reactions reflect a better allocation of investment:Aup

(respectivelyAimp) moves down as the financing conditions are less demanding, which

reduces the distortion at the top (resp. at the bottom). Comparing the optimal measure-

ments with and without moral hazard, write-up measurements aboveAup and impairment

measurements are second best measurements due to the presence of moral hazard. The

distortion due to moral hazard is decreasing in the economy’s wealth. If the economy is

extremely wealthy, there is no financing issue and optimal measurements are in line with

the measurements without moral hazard.

As Proposition 5 shows when the aggregate wealth in the economy is large (E(A) >

W low), the optimal information system can implement first best for low values of the

cost of capital: firms withA ∈ [0, AH ] receive financing and run the project, while firms

with A > AH take the financial investment. In this case write-up measurements resolve

the distortion at the top without altering the financing issue of firms at the bottom. For

higher costs of capital, only firms withA > Aup take the financial investment. Firms

with A ∈ [AH , Aup] take the inefficient investment and there is overproduction. When

the wealth in the economy is belowW imp, impairment measurements becomes optimal

for high values of the cost of capital to squeeze some firms at the bottom to allow the

remaining non disclosing firms to meet the financing requirements. In the impairment

equilibrium, disclosing firms are too risky from the lenders’ perspective while in the write-

up equilibrium, disclosing firms are very safe firms to lend money but forgo voluntarily to

run the project. The distortion on the bottom is only partially resolved as only firms with

A ∈ [Aimp, AL] receive financing and run the projects at the expense of the distortion at

the top which is not alleviated.

Cost of Capital and Optimal Information System

We have just discussed two types of measurement: impairment and write-up. The impairment-

like measurement is similar to the optimal information system in Goex and Wagenhofer

(2009). In their setting, ifE(A) > AL, absent any information all firms take the pro-

ductive investment and there is no distortion. In contrast ifE(A) ≤ AL absent any

information all firms lose their productive investment. There is no alternative investment

and thusAL is independent of the cost of capital. In the paper the financing conditions

are not the only constraint to run the project. The financial investment can deter firms to

take the productive investment and might create new distortions.

22

The write-up measurement occurs as a result of the new tension between productive

and financial investments. Our next result links the optimal information system to the

exogenous cost of capital.

Corollary 1

(i) If γmin < γ < min(max(γL, γmin), γmax), a write-up information system is opti-

mal. There exists a unique cost of capitalγup ∈ (γmin, min(max(γL, γmin), γmax))

such thatAH(γup) = Aup(γup) and:

– If γmin < γ < γup, firms aboveAH measure their assets.

– If γup < γ < min(γL, γmax), firms aboveAup as defined in Lemma 5 measure

their assets.

γup is increasing inp, H, and η, decreasing inI and c. If there is a collateral

squeezeγup increases (decreases) if expression(4) is positive (negative):

AH(γup)

AL(γup).(AH(γup)− AL(γup))

(1− ε)−

∫ AH (γup)

1−ε

0 f(A)dA

f(

AH(γup)

1−ε

)

︸ ︷︷ ︸inverse hazard rate

(4)

(ii) If min(max(γL, γmin), γmax) < γ < γmax an impairment-like information system

is optimal. Firms below the thresholdAimp as defined in Lemma 5 measure their

assets.

(iii) If γ > γmax, firms take the financial investment and are indifferent between any

information system.

In Figure 2 below we provide an example comparing the equilibrium investment

choice without information and with optimal partial measurement. To expose the effect of

regulation relatively to no regulation, we keep fixed all parameters and gradually change

the aggregate wealth in the economy. Recall thatγmax is independent of a reduction of

the assets in the same proportion for all firms. In this exampleγup is increasing in the ag-

gregate wealth. However,γL is increasing inE(A) andγH is decreasing inE(A), which

affects the optimal measurement, the investment choice and distortion.

23

γup γmax < γL

Optimalmeasurement

Write-up measurementaboveAH

Write-up measurementaboveAup

“No information”benchmark

Only financialinvestment

γH = γmin︸ ︷︷ ︸ ︸ ︷︷ ︸First-best investment

for all firms

Overproduction distortion

for A ∈ [AH , Aup]

︷ ︸︸ ︷

Underproduction distortion

for all firms

Case 1: High aggregate wealth (E(A) = Wmax = AH(γmin))(E(A) > AH > AL for γ ∈ (γmin, γmax))

γH γup γmax < γL

Optimalmeasurement

Write-up measurementaboveAH

Write-up measurementaboveAup

“No information”benchmark

Only productiveinvestment

Only financialinvestment

γmin︸ ︷︷ ︸ ︸ ︷︷ ︸First-best investment

for all firms

Overproduction distortion

for A ∈ [AH , Aup]

︷ ︸︸ ︷ ︷ ︸︸ ︷

Overproduction distortion

for all firms

Underproduction distortion

for all firms

Case 2: Above intermediate aggregate wealth (E(A) > AL(γmax) = W imp)(AH > E(A) > AL for γ ∈ (γmin, γH) andE(A) > AH > AL for γ ∈ (γH , γmax))

γup γmax = γL = γH

Optimalmeasurement

Write-up measurementaboveAH

Write-up measurementaboveAup

“No information”benchmark

Only productiveinvestment

γmin︸ ︷︷ ︸ ︸ ︷︷ ︸First-best investment

for all firms

Overproduction distortion

for A ∈ [AH , Aup]

︷ ︸︸ ︷

Overproduction distortion

for all firms

Case 3: Intermediate aggregate wealth (E(A) = W imp = AL(γmax))(AH > E(A) > AL for γ ∈ (γmin, γmax))

24

γLγup γmax < γH

Optimalmeasurement

Write-up measurementaboveAH

Write-up measurementaboveAup

“No information”benchmark

Only productiveinvestment

Only financialinvestment

Impairment measurementbelowAimp

γmin︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸First-best investment

for all firms

Overproduction

for A ∈ [AH , Aup]

︷ ︸︸ ︷︷ ︸︸ ︷

Overproduction distortion

for all firms

Underproduction distortion

for all firms

Overpr. forA > AH

Underpr. forA < Aimp

Case 4: Low aggregate wealth (AL(γmin) = W low < E(A) < W imp = AL(γmax))(AH > E(A) > AL for γ < γL, elseAH > AL > E(A))

γmax < γH

Optimalmeasurement

Impairment measurementbelowAimp

“No information”benchmark

Only financialinvestment

γL ≤ γmin︸ ︷︷ ︸Overproduction distortion for firmsA > AH

Underproduction distortion for firmsA < Aimp

︷ ︸︸ ︷

Underproduction distortion

for all firms

Case 5: Very low aggregate wealth (E(A) ≤W low = AL(γmin))(AH > AL > E(A))

Figure 2: Information system, investment choice and distortion

If the amount of aggregate wealthE(A) ∈ [Wmax,W imp], firms can receive funding

for their project (cases 1, 2 and 3) even without information and impairment measure-

ments are never implemented. It is optimal to prescribe write-up measurement that either

achieves first best or creates overproduction only for a small proportion of the firms (those

with A ∈ [AH , Aup]). From Case 1 to case 3 the wealth decreases and thus the set of write-

up measurements attaining first best shrinks to satisfy the financing requirements for non

disclosing firms. When the aggregate wealth falls belowW imp, lenders require higher

amount of collateral and financing becomes an issue for high costs of capital (in case 4

this happens forγ ∈ [γL, γmax]). If the wealth further drop belowW low, impairment mea-

surements are optimal no matter the cost of capital. Firms with pledgeable assets below

the impairment threshold are squeezed out to allow the rest of the firms to receive financ-

ing. Relatively to first best there will be underproduction for firms below the impairment

threshold and overproduction for firms aboveAH .

25

The underlying force of the optimal accounting measurement rules is the level of ag-

gregate wealth in the economy relatively to the financing conditions that depend on the

cost of capital. The model predicts that a rich economy exhibits write-up measurements

and the level of cost of capital is irrelevant. A poorer economy prescribes respectively

write-up accounting for low costs of capital and impairment measurements for high val-

ues. However first best is attained if and only if write-up measurements are optimal and

the cost of capital is relatively low.

4 Endogenous Cost of Capital

The assumption that capital is in perfectly elastic supply at an exogenous cost of capital is

a simplification. We complete the analysis by recognizing the endogeneity of the supply

and the cost of capital. To do so we assume that the entrepreneurial firms receive capital

from two sources: from an exogenous supplyK > 0 and from firms that do not run the

project.

4.1 Benchmarks

We first define two relevant benchmarks. We first define theconstrained first-bestthat

captures the transfer of capital between entrepreneurial and financing firms. Then we

solve for the endogenous cost of capital in absence of moral hazard to isolate the effects

of moral hazard on the cost of capital and optimal measurements in section 4.2.

Constrained First Best

To describe the constrained first best we introduce the notion of a benevolent social plan-

ner, who observes effort and asset values, and maximizes the aggregate wealth in the

economy by optimally allocating the existing resources. The only limitation that he is

facing is the scarcity of resources. Although the productive investment is economically

viable, the social planner cannot fund all firms. He can finance the productive projects

with the exogenous supplyK and with the assets of firms that do not run the project, i.e.

financing firms. The financing firms can only lend what they own, i.e. their pledgeable as-

sets. In this constrained first-best, a firm with pledgeable assetsA generates a net present

value ofpH + ηA − c if the productive investment is funded, otherwise zero. When de-

ciding which firms should run the productive investment the social planner maximizes the

sum of individual utilities. Proposition 6 summarizes the characteristics of the socially

26

optimal productive investment:

Proposition 6 The social optimal productive investment is:

∫ A∗

0

(pH + ηA− c)f(A)dA

where I

∫ A∗

0

f(A)dA = K +

∫ ∞

A∗

Af(A)dA (5)

The social planner wants to fund as many projects as possible, but the resource con-

straint (5) limits his choice. This constraint reflects the transfer of capital between capital

providers and entrepreneurial firms. In our benchmark firms with pledgeable assets below

A∗ run the productive investment and become entrepreneurial firms whereas firms above

A∗ become financing firms. Our constrained first-best is defined differently relatively to

the partial equilibrium setting. There is no cost of capital as the firms abandoning the

project transfer their assets to the other firms to fill the financing gap. We find that in such

a setting it is optimal for the capital rich firms to finance the capital poorer firms. Given

that running the project requires the same external investment and that capital rich firms

are not endowed with better projects, the transfer of high value assets to the poorer firms is

optimal. The allocation of resources across firms can be better understood by rearranging

condition (10) as follows:

K + E(A)︸ ︷︷ ︸

Resources

=

∫ A∗

0

(I + A)f(A)dA

︸ ︷︷ ︸Invested Capital

(6)

The resources in the economy are equal to the initial aggregate wealth in the econ-

omy from the firms and the outside investors providing capitalK. These resources are

dispatched to the entrepreneurial firms: the entrepreneurial firms keep their initial wealth

A and receive the supplemental outside capitalI to run their project. Since resources

are limited, if external financingI increase or projects are less attractive (lowerp, H and

η or higherc), the overall productive investment in the economy reduces. Similarly, a

collateral squeeze is compensated by more firms lending their assets. As a consequence,

if there is a collateral squeeze, the productive investment decreases leading to less social

surplus. This last result is in sharp contrast with the unconstrained first-best in the partial

equilibrium setting. The partial equilibrium setting was neglecting the offsetting effect of

less wealth on the capital providers, if they are determined endogenously. Less wealth

for the capital providers means that they cannot finance as many entrepreneurial firms

27

as before. The effect of macroeconomic factors on the constrained first-best investment

thresholdA∗ and the total social surplus are summarized in the following Corollary:

Corollary 2

(i) A∗ decreases inI and in a collateral squeeze or credit crunch in the economy.

(ii) The total social surplus increases inK, p, H andη, and decreases inI andc.

In general equilibrium, we can address the effect of a credit crunch. This alternative

shock that tightens capital has similar effects on the aggregate productive investment. The

shrinkage in capital is also compensated by more firms lending their assets. However a

lower exogenous supply affects directly the left hand side of resource constraint (6) with-

out altering directly the right hand side, in contrast to a collateral squeeze. The amount of

capital invested in entrepreneurial firms is indirectly adjusted through the thresholdA∗.

Thus although the net effect of the collateral squeeze or the credit crunch on the aggregate

productive investment is qualitatively the same, their impact on the thresholdA∗ are in

magnitude different. This latter response will explain different outcomes on the cost of

capital in presence of a collateral squeeze or credit crunch in the subsequent analysis.

General Equilibrium without Moral Hazard

Before solving the general equilibrium with moral hazard, we study the general equi-

librium when effort is observable which we will use as an additional benchmark. From

proposition 2, we know that firms aboveAH disclosing their assets and becoming financ-

ing firms and the remaining firms demanding capital is one of the optimal information

system and would be the “least cost” information system should the firms pay a cost to

measure. Hence, the net demand without moral hazard is defined as follows:

I

∫ AH

0

f(A)dA−K −∫ ∞

AH

Af(A)dA (7)

AH is decreasing in the cost of capital and thus the net demand defined above decreases

in the cost of capital. We show the existence of a unique cost of capital that clears the

markets. Not surprisingly without moral hazard, we can achieve the constrained first best.

We will use the cost of capitalγFB as a benchmark when we define the optimal cost of

capital with moral hazard.

Proposition 7 Without moral hazard, there exists a unique general equilibrium that achieves

the constrained first best:

28

(i) Markets clear atγFB such that:

I

∫ AH(γFB)

0

f(A)dA = K +

∫ ∞

AH(γFB)

Af(A)dA (8)

(ii) Firms aboveAH(γFB) = A∗ measure their assets and take the financial investment.

Firms belowAH(γFB) = A∗ run the project

The comparative statics onAH(γFB) are identical toA∗, and differ from our findings

in the partial equilibrium analysis:

Corollary 3

(i) γFB is increasing inp, H and η, and decreasing inc and K. A change inI has

ambiguous effects onγFB.

(ii) AH(γFB) is not affected by changes inp, H, η andc, but is decreasing inI. If there

is a collateral squeeze or credit crunchAH(γFB) decreases.

Except forI, the measurement thresholdAH(γFB) is the only channel through which

the market clearing condition is adjusted. The endogenous cost of capital can perfectly

offset the change in the economic environment on this threshold so that the optimal invest-

ment allocation is preserved. However,AH(γFB) decreases in the outside capital raised.

The direct effect of an increase inI is to move up the demand while a higherI decreases

indirectly the measurement threshold, and drives down the demand. All in all, both credit

crunch and collateral squeeze also call for lowerAH(γFB), achieved through lowerγFB.

The intuition behind this result is that both credit crunch (leading to lower exogenous

supply) and collateral squeeze need to be compensated by more firms lending their assets

and less firms taking the project.

4.2 Moral Hazard and Cost of Capital

In Section 3.3, we have discussed the optimal measurement for a given cost of capital,

i.e. in a partial equilibrium setting. We now derive the general equilibrium by taking into

account the effects of endogenous supply. To this end, we introduce the market clearing

condition, stating that the demand in capital needs to equal the supply of capital. We study

conditions on the cost of capital to clear the markets. We also characterize the possible

general equilibria as a function of the exogenous parameters.

29

Net Demands

In presence of moral hazard, the determination of the cost of capital is more complex. Due

to the the different measurement rules enforced, the net demand (demand net of supply)

is a piecewise function defined by:

Q(γ) = I

∫ AH(γ)

0

f(A)dA−K −∫ ∞

AH(γ)

Af(A)dA, if γ ∈ (γmin, γup],

T (γ) = I

∫ Aup(γ)

0

f(A)dA−K −∫ ∞

Aup(γ)

Af(A)dA, if γ ∈ (γup, min(γL, γmax)],

Z(γ) = I

∫ ∞

Aimp(γ)

f(A)dA−K −∫ Aimp(γ)

0

Af(A)dA, if γ ∈ (min(γL, γmax), γmax).

The net demandsQ(γ) andT (γ) are associated with write-up measurements. In both

cases the entrepreneurial firms are the ones “at the bottom” and the financing firms are

the ones “at the top”. The net demand inZ(γ) corresponds to impairment-like measure-

ment in which case the entrepreneurial firms are the ones “at the top” and the financing

firms are the ones “at the bottom”. Although net demandsQ(γ) andZ(γ) differ in their

measurement rules, they both decrease in the cost of capital and are affected similarly

by a change in the exogenous parameters and in particular inI. On the demand side,

higher outside capital shrinks the proportion of firms financed while increasing the per

firm outside capital. On the supply side, higher outside capital unambiguously increases

the supply as it makes the financing requirements tighter. In this case, the shift of the net

demand is intuitive. In contrast, the net demandT (γ) moves in the opposite direction.

Counter-intuitively, it increases in the cost of capital and also in the outside capital raised.

Lemma 5

(i) If IAL(γmax)

≥ K+∫ Aimp(γmax)

0 Af(A)dA∫∞Aimp(γmax) Af(A)dA

, impairment-like measurements are never opti-

mal.

(ii) If IAL(γup)

≥K+

∫∞AH (γup) Af(A)dA

∫ Aup(γup)0 Af(A)dA

or A∗ < AH(γup), write-up measurements are

never optimal.

Lemma 5 outlines sufficient conditions to rule out measurements in general equilib-

rium. If the above conditions are met, there is excess demand. Excess demand emerges

if capital providers’ wealth relatively to entrepreneurial firms’ wealth is lower than their

revenue shortfall in the worst case scenario, i.e., when the project fails and they get pos-

session of the minimum collateral. This is more likely to happen if the amount of outside

30

capital to finance the project is relatively high or if the minimum collateral to guarantee

financing is low and/or if the exogenous capital is low. The ratioIAL(γ)

evaluated atγup

or γmax can also be interpreted as a measure of the riskiness of the project or a solvency

ratio. The higher the ratio, the more risky the project is. Alternatively this ratio cap-

tures the outside financing relatively to internal financing. To guarantee the existence of

impairment-like measurements or write-up measurements, we need more structure than in

the partial equilibrium setting. Lenders’ assets need to be higher than the revenue short-

fall that they can incur.The solvency ratio is higher for write-up measurements compared

to impairment-like measurements. In other words capital providers do not need to be as

wealthy so that impairment-like measurements arise in equilibrium. A sufficient condi-

tion to rule out write-up measurements in general equilibria is to have the constrained

first-best thresholdA∗ lower than the write-up threshold where the two types of write-up

measurements meetAup(γup) = AH(γup) .18 If this is the case there will be not enough

firms funding the entrepreneurial firms and there will be excess demand.19

General Equilibria

If one or both of the conditions described in Lemma 5 are not satisfied, there exists at least

one optimal cost of capital that clears the markets. We next derive the general equilibria

as a function of the aggregate wealth in the economy:

Proposition 8

- If IAL(γup)

<K+

∫∞AH (γup) Af(A)dA

∫ Aup(γup)0 Af(A)dA

andE(A) > W low, there are two general equilib-

ria:

GE1: the optimal rule prescribes firms to measure assets aboveAH(γ∗) and the

optimal cost of capitalγ∗ = γFB < γup achieves first-best and is defined by

I∫ AH(γ∗)

0f(A)dA = K +

∫∞AH(γ∗)

Af(A)dA.

GE2: the optimal rule prescribes firms to measure assets aboveAup(γ∗∗) > AH(γ∗∗)

and the optimal cost of capitalγup < γ∗∗ < γmax achieves first-best and is

defined byI∫ Aup(γ∗∗)

0f(A)dA = K +

∫∞Aup(γ∗∗)

Af(A)dA.

18Given thatAH(γ) is decreasing inγ andAup is increasing inγ the lowest possible write-up thresholdis AH(γup) = Aup(γup).

19However if there is always excess demand forγ < γmax, there exits an equilibrium atγmax, where thenon-disclosing firms are indifferent between the productive and the financial investment. This equilibriumis stable.

31

- If IAL(γmax)

≥ K+∫ Aimp(γmax)

0 Af(A)dA∫∞Aimp(γmax) Af(A)dA

andW low < E(A) ≤ W imp, there is an equi-

librium GE3 defined by:

GE3: the optimal rule prescribes firms to measure assets belowAimp(γ∗∗∗) and the

optimal cost of capitalγL < γ∗∗∗ < γmax is defined byI∫∞

Aimp(γ∗∗∗)f(A)dA =

K +∫ Aimp(γ∗∗∗)

0Af(A)dA.

- If IAL(γmax)

≥ K+∫ Aimp(γmax)

0 Af(A)dA∫∞Aimp(γmax) Af(A)dA

, IAL(γmin)

≤ K+∫ Aimp(γmin)

0 Af(A)dA∫∞Aimp(γmin) Af(A)dA

andE(A) ≤

W low, there is a unique general equilibriumGE3.

In all three general equilibria disclosing firms invest in the open market while non

disclosing firms run the project.

The lowest feasible cost of capitalγ∗ clears net demandQ(γ) and is associated with

liquidation of capital rich firms. When it exists,γ∗ is equal toγFB. The intermediate

cost of capitalγ∗∗ clears net demandT (γ) and also attains first best. The highest feasible

cost of capitalγ∗∗∗ clearsZ(γ) and is associated with measurement of capital poor firms.

The latter cost of capital introduces distortions in the productive investment, because both

capital poor firms and capital rich firms are selecting a suboptimal investment.20

Corollary 4

(i) If γmin < γup < γmax < γL and ifA∗ > AH(γup), there are two general equilibria,

described byGE1 andGE2.

(ii) Assumeγmin < γup < γL < γmax. Then,

– if A∗ > AH(γup) and I∫∞

Aimp(γmax)f(A)dA −

∫ Aimp(γmax)

0Af(A)dA < 0,

there are three general equilibria, described byGE1, GE2 andGE3.

– if A∗ > AH(γup) and I∫∞

Aimp(γmax)f(A)dA −

∫ Aimp(γmax)

0Af(A)dA > 0,

there are two general equilibria, described byGE1 andGE2.

– if A∗ < AH(γup) and I∫∞

Aimp(γmax)f(A)dA −

∫ Aimp(γmax)

0Af(A)dA < 0,

there is one general equilibrium, described byGE3.

(iii) If γL < γmin < γmax andI∫∞

Aimp(γmax)f(A)dA−

∫ Aimp(γmax)

0Af(A)dA < 0, there

is one general equilibrium described byGE3.

20At γ∗, the first best for a given cost of capital and the first best in the economy with limited resourcesare equal. Atγ∗∗, the constrained first best is attained but not the unconstrained first best. However atγ∗∗∗,A∗ > AH and hence, there is a distortion relative to both types of first best .

32

In Figure 3 below we pursue the analysis of the five cases illustrated in Figure 2 and

present the possible general equilibria when the conditions in the economy ensure market

clearing.

I −K = 0.10

−0.54

-

6

1 1.321.18

Net demand

γ

γ∗ = 1.12; γ∗∗ = 1.30

γmin γmaxγupγ∗ γ∗∗

︸ ︷︷ ︸ ︸ ︷︷ ︸Write-up aboveAH Write-up aboveAup

Figure 3a: High aggregate wealth (ε = 0)

I −K = 0.10

−0.10

−0.34

-

6

1 1.321.18

Net demand

γ

γmin γup γL γmaxγ∗ γ∗∗ γ∗∗∗

1.22

︸ ︷︷ ︸ ︸ ︷︷ ︸︸ ︷︷ ︸Write-up aboveAH Write-up aboveAup Impairment belowAimp

γ∗ = 1.15; γ∗∗ = 1.19; γ∗∗∗ = 1.24

Figure 3b: Low aggregate wealth (ε = 0.23)

33

0.08

-

6

1

γmin

1.32

γmax

Net demand

γ

γ∗∗∗ = 1.01

γ∗∗∗

︸ ︷︷ ︸Impairment belowAimp

Net demand

Figure 3c: Very low aggregate wealth (ε = 0.67)

Figure 3: Example of General Equilibrium as a Function of Aggregate Wealth

p = 12 ; Δp = 3

32 ; c = 14 ; H = 6; I = 2; K = 1.9; f(A) = λe−λA; λ = 1; E(A) = 1; η = 1

4.3 Macroeconomic Shocks & General Equilibrium Impact

We analyze the effects of macroeconomic factors on the equilibrium costs of capital and

the measurement thresholds with moral hazard. Macroeconomic shocks can effect all the

exogenous parameters in the economy. We study first the potential impacts of macroe-

conomic shocks on the characteristics of the productive investment, then investigate how

the different players could rather coordinate towards one equilibrium rather than another

one and finally study the effects of capital tightening shocks, namely credit crunch and

collateral squeeze on the costs of capital and asset measurement rules.

Changes in the characteristics of the productive investment

The net demands determiningγ∗ andγ∗∗∗ are affected similarly by changes in the funda-

mentals of the productive investment. Intuitively more returns on the project (higherp, H

or η) or less moral hazard (lowerc) translate into more demand in capital. A higher cost

of capital can counterbalance this surge in demand so that the market clearing condition

is preserved. If the outside capitalI increases, entrepreneurial firms need easier financing

34

conditions, a decrease of the cost of capital fulfills that goal for impairment measurement.

For write-up measurements the increase inI depends on the characteristics of the distri-

bution. On the contrary, the net demand related toγ∗∗, is atypical and shifts in an opposite

direction to the other net demands after a change in a parameter of the economy. As a

result, a better project, less severe moral hazard or lower outside capital raised decrease

the demand and a lower cost of capital would counterintuitively worsen the excess sup-

ply. To address an excess supply the cost of capital in equilibrium moves upward. The

costs of capitalγ∗∗ andγ∗∗∗ are similarly affected byΔp, while the cost of capitalγ∗ is

independent ofΔp. This feature is consistent with the cost of capitalγ∗ being the only

equilibrium cost of capital in line with the equilibrium cost of capital in absence of moral

hazardγFB. The comparative statics of the equilibrium costs of capital are summarized

below:

Corollary 5

(i) γ∗, γ∗∗ andγ∗∗∗ are increasing inp andH, and decreasing inc. γ∗ is unaffected

by a change inΔp and I has ambiguous effects onγFB, while γ∗∗ and γ∗∗∗ are

decreasing inΔp andI.

(ii) AH(γ∗), Aimp(γ∗∗∗) andAup(γ

∗∗) do not change in response to a change inp, c,

Δp, η or H. An increase inI increasesAimp(γ∗∗∗) and decreasesAH(γ∗) and

Aup(γ∗∗).

(iii) AL(γ∗), AL(γ∗∗) andAL(γ∗∗∗) are unaffected byp, c, Δp, η or H. An increase in

I increasesAL(γ∗∗∗) and decreasesAL(γ∗) andAL(γ∗∗).

In contrast to the partial equilibrium, the equilibrium thresholds are not affected in

response to a change inp, Δp, η or H. An increase in these parameters is exactly offset

by an increase in the cost of capital so that the market clearing condition is preserved.

Similarly, a lower cost of capitalc charged counterbalances the increase inc, which leaves

the thresholds unaffected. An increase inI increasesAimp, which is the same effect as in

partial equilibrium and counterintuitively decreasesAup . In response to a raise inI, the

cost of capital decreases andAup moves downward. The maximum collateral to prefer

productive investmentAH when it defines the optimal measurements does not adjust to

any change in the parameters in the economy except the amount of raised capitalI, which

ambiguously moves it either upward if it increases. The feedback effects on the minimum

collateral required to obtain financingAL are in correspondence with the thresholdsAimp,

Aup or AH depending on the general equilibrium selected. In particular during a recession

35

p andH are more likely to decrease and the minimum collateral to find financingAL

remains the same as the equilibrium cost of capital increases. In other terms if the change

is local and we stay in the same general equilibrium, the measurement thresholds remain

the same and all the shocks onp or H are absorbed by the change in cost of capital.

If the aggregate wealth is sufficiently large, the moral hazard can be resolved by the

markets and efficient investment is implemented (cases 1, 2 and 3). There exist two equi-

librium costs of capital,γ∗ = γFB andγ∗∗, andAH(γ∗) = Aup(γ∗∗) = A∗ . A decrease

in the aggregate wealth (case 3) adds the inefficient equilibriumGE3. In this scenario, the

cost of capitalγ∗∗∗ is always higher than the cost of capital without moral hazardγFB. A

further drop in aggregate wealth leaves the economy with only the inefficient equilibrium.

As illustrated in cases 4 and 5, we move from an economy with three general equilibria

to an economy with a unique equilibrium without being confronted with an intermedi-

ate economy facing two equilibria. All in all a reduction in aggregate wealth creates

inefficient investment. The model predicts that when there is multiplicity of equilibria

in an economic environment with the same aggregate wealth we could observe diamet-

rically opposite measurements. The model could capture how fair value accounting is

related to the cost of capital. An economy with the same wealth could feature two types

of measurement: if financing requirements are loose, firms with fair value accounting

would recognize gains whereas if the financing conditions are tighter, they would rec-

ognize losses. With the same wealth, an economy with a low cost of capital is more

informative about high type events, whereas an economy with a high cost of capital is

more informative about low type events. This would mean that recognition of losses and

high cost of capital is not necessarily a sign of a decrease in assets’ values.

Alternatively, one could interpret the multiplicity of equilibria by the existence of

different measurement requirements for different type of events. On one hand, incurred

administrative or marketing costs are expensed immediately as they cannot be linked to

the expected benefits. This is an example for measurement of bad news. On the other

hand, certain types of costs such as the costs to produce inventory for sale, research and

development costs21 or borrowing costs of qualifying assets (formerly FAS 34) for US

GAAP and IAS 23Borrowing Costsfor IFRS can be capitalized when it is probable that

they will result in future economic benefits to the entity, which is a measurement ofgood

21Under IFRS development costs are capitalized when technical and economic feasibility of a projectcan be demonstrated in accordance with specific criteria. Some of the stated criteria include: demonstratingtechnical feasibility, intent to complete the asset, and ability to sell the asset in the future, as well as others.Under IFRS this is done only for computer software development costs under ASC 985-20 and ASC 350-40.

36

news.22

Lastly, one can think of the co-existense of IFRS and US GAAP as two different mea-

surement requirements under two general equilibria. Generally speaking IFRS allows for

more write-up type measurements, while US GAAP allows for more impairment type

measurements. For example, IFRS allows for all types of development costs to be cap-

italized as long as there is economic and technical feasibility of the project, while US

GAAP allows this only for computer software developed for external use. Another ex-

ample is that IFRS allows upward revaluation of assets, while US GAAP does not permit

that. There are two ways to interpret the current co-existence of these two measurement

requirements. First, one can claim that difference in aggregate wealth, exogenous supply

or other economy parameters between US and the countries adopting the international

standards may have led to two separate equilibria. Second, in a global environment the

difference between the macroeconomic factors of different countries might not be sig-

nificant. If this is so, the co-existence of IFRS and US GAAP can be interpreted as

multiplicity of equilibria (case 4 in our descriptive examples in Figure 4).

Stability of the General Equilibria

In the preceding analysis we have shown that multiplicity of equilibria may arise. The

common issue is to know how agents in the economy will coordinate towards one equilib-

rium rather than another one. To better understand the dynamics at play for converging to-

wards one equilibrium over another, we introduce the concept of Walrasian tatonnement.

The tatonnement process is a model for investigating stability of equilibria, under which

no transactions and no production take place at disequilibrium prices. Instead, the cost

of capital decreases (increases) if there is excess supply (demand). This concept clarifies

how players in the economy would react to a small shock, potentially due to macroeco-

nomic fundamentals.

Corollary 6 There are only two stable general equilibria:GE1 andGE3.

The counterintuitive features of the net demand associated withγ∗∗ violate the taton-

nement process and the cost of capitalγ∗∗ is ruled out. On one hand, the lowest costof

22Borrowing costs may include interest on banks overdrafts and short and long term borrowings; amorti-zation of discounts and premiums relating to borrowings; amortization of ancillary costs incurred in connec-tion with the arrangement of borrowings etc. An example of such assets is inventories that need substantialtime to bring them to their saleable condition. Borrowing costs that are directly attributable to the acquisi-tion, construction or production of a qualifying asset shall be capitalized as part of the cost of the asset asper ASC 835-20Capitalization of Interest.

37

capitalγ∗ = γFB, if exists, is stable and associated with write-up measurements and effi-

cient investment. At this cost of capital, the constrained first best coincides with the first

best when the cost of capital is taken as exogenous. On the other hand, the highest cost of

capitalγ∗∗∗, if it exists, is also stable, but is associated with inefficient investment. This

cost of capital elicits all firms “at the top” to run the project, which is suboptimal, while

forcing firms “at the bottom” to take the financial investment, which is not efficient. How-

ever this concept does not select a unique equilibrium. Selecting the write-up equilibrium

over the impairment equilibrium on the grounds of Pareto efficiency is not a good criteria

as experiments have shown that players do not necessarily coordinate on the Pareto effi-

cient equilibrium.23 The multiplicity of equilibria could also be seen as a richness as a

change in aggregate wealth could lead from an economy with one stable equilibrium to

an economy with two stable equilibria. Although the two equilibria are stable, it is hard to

coordinate all the players to the other type of equilibrium. All the players have to change

their anticipations to “adopt” the alternative equilibrium. This question of coordination

especially arises when capital tightening shocks decrease the aggregate wealth. If the re-

duction in aggregate wealth is sufficiently large, we might switch from an economy with

one stable equilibrium to an economy with two stable equilibria as illustrated in Figure 4.

Effect of Capital Tightening

We further analyzes the impact of macroeconomic shocks changing the net demand. We

derive the effect of a collateral squeeze and a credit crunch if these shocks are local shocks.

Corollary 7

If there is a collateral squeeze,

(i) γ∗ increases, whileγ∗∗ andγ∗∗∗ decrease.

(ii) AH(γ∗) andAup(γ∗∗) decrease, whileAimp(γ

∗∗∗) increases.

(iii) AL(γ∗) andAL(γ∗∗∗) increase whileAL(γ∗∗) decreases.

If there is a credit crunch,

(i) γ∗ andγ∗∗∗ increase, whileγ∗∗ decreases.

(ii) AH(γ∗) andAup(γ∗∗) decrease, whileAimp(γ

∗∗∗) increases.

23Devetag and Ortmann (2007) review findings in the experimental literature on coordination failures andin particular when subjects fail to coordinate on the Pareto efficient equilibrium. Bloomfield and Kadiyali(2005) also show how subjects might select an equilibrium, that is ruled out by Cho-Kreps intuitive criteriaand give support to the co-existence of multiple equilibria.

38

(iii) AL(γ∗) andAL(γ∗∗∗) increase whileAL(γ∗∗) decreases.

If there is a small collateral squeeze, the aggregate wealth decreases and the two stable

equilibria move in opposite directions. Although their associated net demands are affected

similarly by a change in cost of capital, they adapt differently to a collateral squeeze.

For the “first best” equilibrium the excess demand is offset by a higher cost of capital

and more firms take the financial investment, while for the “impairment” equilibrium as

the financing conditions are tighter a lower cost of capital partially compensates for the

decrease in wealth. This decrease in the cost of capital is not sufficient and more firms

at the bottom take the inefficient investment short of financing. The write-up unstable

equilibrium decreases in response to excess demand. The effect of the collateral squeeze

moves the allocation of investment for write-up measurements in the opposite direction

relatively to the partial equilibrium analysis.

I −K = 0.10

−0.10

−0.16

−0.34

−0.28

-

6

1

γmin γmax

1.32

Net demand

γ

γ∗ γ∗∗ γ∗∗∗←←→

....................................

......

...........................

....ε = 0.23, γ∗ = 1.146, γ∗∗ = 1.191, γ∗∗∗ = 1.238

ε = 0.20, γ∗ = 1.141, γ∗∗ = 1.205, γ∗∗∗ = 1.254

Figure 4: Example of the Effect of Collateral Squeeze on GE

p = 12 ; Δp = 3

32 ; c = 14 ; H = 6; I = 2; K = 1.9; f(A) = λe−λA; λ = 1; E(A) = 1; η = 1

39

I −K1 = 0.15

I −K0 = 0.10

−0.10

−0.07

−0.34

−0.31

-

6

1

γmin

1.32

γmax

Net demand

γ

γ∗ γ∗∗ γ∗∗∗→←→

....K1 = 1.85, γ∗ = 1.16, γ∗∗ = 1.18, γ∗∗∗ = 1.25

............................... . .. . .

. . .. . .

.................

K0 = 1.90, γ∗ = 1.15, γ∗∗ = 1.19, γ∗∗∗ = 1.24

Figure 5: Example of the Effect of Credit Crunch on GE

p = 12 ; Δp = 3

32 ; c = 14 ; H = 6; I = 2; ε = 0.23; f(A) = λe−λA; λ = 1; E(A) = 1; η = 1

The nature of the two shocks affect differently the financing constraint and the net

demand: a collateral squeeze directly affects simultaneously the financing constraint and

the net demand, while a credit crunch only directly changes the net demand. A credit

crunch and a collateral squeeze affect similarly the “write-up” equilibria. However, the

effect of a credit crunch on the impairment equilibrium is reversed compared to the col-

lateral squeeze. With a credit crunch the equilibrium cost of capital increases while it

decreases with a collateral squeeze. This difference is related to the fact that the collateral

squeeze shifts the maximum net demand with impairment measurements to the left and

is attained at a lower cost of capital. In contrast a credit crunch moves the net demand

upward without altering the costs of capital where the minimum and maximum net de-

mand are attained. We illustrate these effects in Figures 4 and 5. Although the net demand

shift under the two types of shocks is not identical, the effect on the aggregate investment

allocation is similar. Both credit crunch and collateral squeeze deter firms from taking the

productive investment and benefit the financial investment. These findings could provide

some rational in explaining the decrease in the short term interest rate during recession -

the impairment equilibrium moves the cost of capital downward with a collateral squeeze.

To summarize, although capital tightening shocks move the impairment cost of capital

in the opposite direction and the write-up costs of capital in the same direction, they both

decrease the productive investment.

40

5 Conclusion

This paper relates optimal measurement policies to the cost of capital when resources

in the economy are limited. Depending on the economic environment different optimal

asset measurements can be implemented. A wealthy economy features write-up mea-

surements that achieve first best, while impairment like measurements might arise in a

less rich economy. The latter measurements are associated with inefficient investment.

Capital poor firms find financing at the expense of capital rich firms which short of infor-

mation take the productive investment. Changes in macroeconomic fundamentals shape

the supply and demand of capital markets, affecting simultaneously measurement rules

and the cost of capital. We study more specifically the effect of a collateral squeeze and a

credit crunch. Their effects on the impairment cost of capital differ but have similar con-

sequences on the investment allocation. In the impairment equilibrium a small decrease

in the value of the pledgeable assets similar to a small negative collateral shock or a small

decrease in the amount of capital squeezes collateral-poor firms out of the credit market

while attracting more capital rich firms to invest in the open market in the write-up equi-

librium. These macroeconomic shocks might also lead to different optimal measurements

and equilibria if their magnitude is relatively large. Thus a very wealthy economy can

switch from an economy with write-up measurements and no investment distortions to an

economy with possibly impairment measurements and investment inefficiencies.

Depending on the optimal measurements that prevails in equilibrium, negative macroe-

conomic shocks can worsen investment inefficiencies. The model suggests that the regu-

latory environment can mitigate inefficiencies by designing more write-up measurements

and help the players coordinating on the efficient equilibrium. The model offers new

insight on the relation between asset measurements and cost of capital in a general equi-

librium setting. The paper isolates the effects of the credit market and cost of capital in a

static model and homogenous lenders. Future research could further expand our analysis

in a dynamic general equilibrium environment or provide a deeper analysis on the role of

banking intermediation.

41

Appendix

Proof of Proposition 1: Part(i): The firm with pledgeable assetsAH is indifferent between run-

ning the productive investment and taking the financial investment.AH is defined as follows:

pH − γI + (ηp + (1− p))AH − c = γAH . Solving forAH yieldsAH = pH−γI−cγ−(ηp+(1−p)) > 0.

Part(ii): The comparative statics are∂AH∂p = H

γ−(ηp+(1−p)) > 0, ∂AH∂H = p

γ−(ηp+(1−p)) > 0,∂AH∂I = −γ

γ−(ηp+(1−p)) < 0, ∂AH∂c = −1

γ−(ηp+(1−p)) < 0, ∂AH∂γ = (1+p(1−η))I−(pH−c)

(γ−(ηp+(1−p)))2< 0,

∂AH∂η = p(pH−γI−c)

(γ−(ηp+(1−p)))2> 0.

Part(iii): The exogenous shock causes a decrease in asset values byε > 0 andpH−γI+(ηp+(1−

p))(1 − ε)AshockH − c = γ(1− ε)Ashock

H . Rearranging,AshockH = 1

1−εpH−γI−c

γ−(ηp+(1−p)) = 11−εAH >

AH .

Proof of Proposition 2: The regulator solves the following optimization program:

maxθ(A)

U(θ(A)) =∫ AH

0(pH − γI + (ηp + (1− p))A− c)(1− θ(A))f(A)dA

+∫ ∞

AH

γA(1− θ(A))f(A)dA

+∫ ∞

0(pH − γI + (ηp + (1− p))A− c)θ(A)f(A)dA

Taking the first order condition (FOC) yields:

∂U(θ(A))∂θ(A)

=

{f(A)((pH − γI + (ηp + (1− p))A− c)− γA) < 0 if A > AH

0 otherwise,

The solution isθ(A) = 0 for A > AH . Otherwise anyθ(A) can be set. As a result, firms withasset valuesA > AH measure and run the productive investment, while the rest of the firms take

the financial investment, which is the unconstrained first-best result.

Proof of Lemma 1: We know thatAL = 1ηp+(1−p)(

pcΔp + γI − pH). The comparative

statics are∂AL∂H = −p

(ηp+(1−p)) < 0, ∂AL∂I = γ

(ηp+(1−p)) > 0, ∂AL∂c = p

Δp(ηp+(1−p)) > 0,∂AL∂γ = I

(ηp+(1−p)) > 0, ∂AL∂η = −p(pc/Δp−pH+γI)

(ηp+(1−p))2< 0, ∂AL

∂Δp = − pcΔp2(ηp+(1−p))

< 0, ∂AL∂p =

c−ΔpH−Δp(η−1)γIΔp(ηp+(1−p)) < 0 aspH − c > (p−Δp)H impliesc−ΔpH < 0.

After a collateral squeeze, the financing condition becomes(1−ε)A ≥ 1ηp+(1−p)

(pcΔp + γI − pH

).

Rearranging,A ≥ AshockL = 1

(1−ε)AL > AL.

Proof of Proposition 3: Let us denote the difference betweenAH(γ) andAL(γ) by J(γ) =pH−γI−c

γ−(ηp+(1−p)) −1

ηp+(1−p)

(pcΔp + γI − pH

).

If γ → γmin, AH →∞ andAL → 1ηp+(1−p)

(pcΔp − pH

)+ I andJ(γ) > 0.

If γ → ∞, AH → 0 and AL → +∞ then J(γ) < 0. Thus there exists aγmax such that

AL(γmax) = AH(γmax). Since∂J(γ)∂γ = ∂AH(γ)

∂γ − ∂AL(γ)∂γ < 0 it follows thatγmax is unique.

Applying the Implicit Function Theorem,∂γmax

∂q = −∂J(γmax)

∂q∂J(γmax)

∂γ

= −∂AH (γmax)

∂q−

∂AL(γmax)

∂q∂AH (γmax)

∂γ−

∂AL(γmax)

∂γ

, where

42

q = H, c, I, η, Δp, p. Hence,∂γmax

∂H > 0, ∂γmax

∂c < 0, ∂γmax

∂I < 0, ∂γmax

∂η > 0, ∂γmax

∂Δp > 0,∂γmax

∂p > 0. The fact thatγmax is not affected by collateral squeeze follows from the fact that

AshockH = 1

(1−ε)AH andAshockL = 1

(1−ε)AL.

If γ ≤ γmax thenAL < AH and it is optimal for firms withAL < A < AH to choose the

productive investment and the remaining firms to select the financial investment. Ifγ > γmax,

AL > AH and the financial investment is always selected. The amount of projects is represented

by F (AH)−F (AL). The derivative ofF (AH)−F (AL) with respect to any exogenous parameter

is of the formA′

Hf(AH) − A′

Lf(AL), whereA′H andA′

L are the derivatives with respect to the

exogenous parameter. Given that whenAH is increasing/decreasing in any of the exogenous

parameters,AL moves in the opposite direction, the comparative statics ofF (AH) − F (AL)

follows immediately from the comparative statics ofAL(γ) andAH(γ). For Δp, A′H = 0 and

A′L < 0, henceF (AH)− F (AL) is increasing inΔp.

Proof of Lemma 2: Part (i): By Proposition 3 firms with assetsA < AL cannot find financing

and do not run the project. Hence, there is an underproduction distortion for those firms. In case of

collateral squeeze the financing threshold increases (by Proposition 2,AshockL = 1

(1−ε)AL > AL)

and the proportion of firms that cannot find financing and do not run the project increases. Hence,

the underproduction distortion worsens.

Part (ii): If AL < AH (i.e if γ < γmax) an information system that does not measure infor-

mation for firms with asset valueA ∈ [AL − δ,AH ] whereδ is small, will alleviate the under-

investment distortion for firms withA ∈ [AL − δ,AL], because the latter are pooled with firms

A ∈ [AL, AH ] and can find financing to run their project. As a result, the proportion of firms run-

ning the project increases. However, ifAL > AH (i.e if γ > γmax) all firms prefer the financial

investment even if they can find financing. In this case the underinvestment distortion cannot be

alleviated.

Proof of Proposition 4: Parts (i) and (ii): We want to determine the cost of capitalγL that

makes the financing thresholdAL equal toE(A), i.e.E(A) = AL(γL) = 1ηp+(1−p)

(pcΔp + γLI − pH

).

Rearranging the expression yields the expression forγL. Similarly, we determineγH as the

cost of capital that makes the investment choiceAH equal toE(A), i.e. E(A) = AH(γH) =pH−γHI−c

γH−(ηp+(1−p)) . Rearranging the expression yields the the expression forγH .

Whenγ ≤ min(γL, γmax, γH) thenAH(γ) ≥ E(A) ≥ AL(γ) and firms prefer the productive

investment (becauseAH(γ) ≥ E(A)) and can find financing (becauseE(A) ≥ AL(γ)).24 Hence,

if γ ≤ min(γL, γmax, γH) and absent any information all firms run the productive investment.

Otherwise, all firms run the financial investment either becauseγ > γH (i.e. E(A) > AH ) and

they prefer the financial investment or becauseγ > γL (i.e. E(A) < AL) and they cannot find

financing for the project without information.

Part (iii): Noting thatΔpH − c > 0 (becausepH − c − γI > 0 and (p − Δp)H −

24Note that whenγL = γmax thenAH(γmax) = AL(γmax) = AL(γL) = E(A) = AH(γH). ThuswhenγL = γmax, it must be the case thatγL = γH .

43

γI < 0 by assumption),∂γL∂H = p

I > 0, ∂γL∂c = − p

ΔpI < 0, ∂γL∂η = E(A)p

I > 0, ∂γL∂I =

−E(A)Δp(ηp+(1−p))+p(ΔpH−c)ΔpI2 < 0, ∂γL

∂p = E(A)Δp(η−1)+(ΔpH−c)ΔpI > 0, ∂γL

∂E(A) = ηp+(1−p)I > 0,

∂γL∂Δp = −E(A)(ηp+(1−p))

ΔpI

(1

Δp − 1)− pc

Δp2I< 0. Further, the comparative statics ofγH is ∂γH

∂H =p

E(A)+I > 0, ∂γH∂c = − 1

E(A)+I < 0, ∂γH∂η = E(A)p

E(A)+I > 0, ∂γH∂I = −E(A)(ηp+(1−p))+pH−c

(E(A)+I)2< 0,

∂γH∂p = E(A)(η−1)+H

E(A)+I > 0, ∂γH∂Δp = 0, ∂γH

∂E(A) = −(ηp+(1−p))(E(A)+I)2

(pH − c− (ηp + (1− p))I)︸ ︷︷ ︸>pH−c−γI>0 by assumption

< 0.

Proof of Lemma 3: Part (i): By Proposition 4 whenγ ≤ min(γL, γmax, γH) all firms run

the productive investment and hence, there is overinvestment distortion. Otherwise, all firms run

the financial investment and hence, there is underproduction distortion.

Part (ii): Let assume that initially before the collateral squeeze the cost of capital isγ ≤

min(γL(W I), γH(W I), γmax) so that at the lockstep all firms run the productive investment. If

the aggregate wealth after the collateral squeezeW shock falls belowW imp andγ ≥ γL(W shock)

none of the firms can find financing, becauseW shock < AL. As a result none of the firms runs the

project and all firms take the financial investment. The economy switches from overproduction to

underproduction. However, ifγ > γL(W shock) firms still prefer the productive investment.

Let now assume that initially before the collateral squeeze the cost of capital is such that

γH(E(A)) ≤ γ ≤ min(γL(E(A)), γmax) so that at the lockstep all firms take the financial

investment becauseE(A) > AH . If the aggregate wealth drops so thatγ ≤ (γH(W shock), then

W shock < AH and firms prefer the productive investment. If at the same timeγ ≤ min(γL(W shock), γmax)

(which can only happen ifW shock > W low), thenW shock < AL and firms can find financing for

their project. As a result all firms take the productive investment. The economy switches from

underproduction to overproduction.

Part (iii): If E(A) < AL < AH none of the firms can find financing and all firms take the

financial investment. Information system that measuresA ∈ [AL, AH ] allows the disclosing firms

to find financing and run the project and hence alleviates partially the distortion on the productive

investment. IfAL < E(A) < AH all firms run the productive investment. An information system

disclosingA > AH dominates no measurement as the disclosing firms will prefer the financial

investment and this alleviate the overproduction distortion.

Proof of Lemma 4: If the non disclosing firms are prescribed to take the financial investment

the regulator solves the following maximization problem:

maxθ(A)

U(θ(A)) =∫ AL

0γA(1− θ(A))f(A)dA +

∫ ∞

AH

γA(1− θ(A))f(A)dA

+∫ AH

AL

(pH − γI + (ηp + (1− p))A− c)(1− θ(A))f(A)dA

+∫ ∞

0γAθ(A)f(A)dA

Taking the first order condition (FOC) yields:

44

∂U(θ(A))∂θ(A)

=

{f(A)(γA− (pH − γI + (ηp + (1− p))A− c)) < 0 if A ∈ (AL, AH)

0 otherwise,

The solution isθ(A) = 0 for AL < A < AH . Otherwise anyθ(A) can be set. As a result,firms with asset valuesAL < A < AH measure and run the productive investment, while the

rest of the firms take the financial investment. This leads to the same distribution of productive

and financial investments across firms as with full measurement and as we have proved before this

is not optimal. It follows that it is never optimal to encourage non-disclosing firms to take the

financial investment.

Proof of Proposition 5: The Lagrangian isL(θ(A)) = U(θ(A))+μ(∫∞

0 (A−AL)θ(A)f(A)dA .

The FOC yields:

For AL < A < AH ,∂L(θ(A))

∂θ(A)= μ(A−AL)f(A) ≥ 0

Else,∂L(θ(A))

∂θ(A)= (pH − γI + (ηp + (1− p))A− c− γA) + μ(A−AL))f(A) (9)

The sign of the FOC is ambiguous, because:

if A < AL, pH − γI + (ηp + (1− p))A− c− γA > 0 andμ(A−AL) < 0

if A > AH , pH − γI + (ηp + (1− p))A− c− γA < 0 andμ(A−AL) > 0

We study the expression (9):

(i) For μ > γ − (ηp + (1− p)), it is increasing inA

(ii) For μ = γ − (ηp + (1− p)), it is flat.

(iii) For μ < γ − (ηp + (1− p)), it is decreasing inA.

Hence, we have four cases to consider to determine the optimal measurement.

CASE 1: μ = 0

If the constraint is not binding thenμ = 0.

(i) For A < AL, ∂L(θ(A))∂θ(A) > 0 andθ(A) = 1.

(ii) For AL < A < AH , ∂L(θ(A))∂θ(A) = 0 andθ(A) can be either0 or 1.

(iii) For For A > AH , ∂L(θ(A))∂θ(A) < 0 andθ(A) = 0.

We can achieve first-best in this case by prescribing no measurement forAL < A < AH . To

summarize, this case returns optimal measurements if∫ AH

0 Af(A)dA/∫ AH

0 f(A)dA > AL and

prescribes firms aboveAH to measure and the others not to measure.

CASE 2: μ > γ − (ηp + (1− p))

When the constraint is binding andμ > γ − (ηp + (1− p)), there exists a uniqueAimp < AL

such that:

45

(i) For A < Aimp, pH − γI +(ηp+(1− p))A− c− γA+μ(A−AL) < 0 and∂L(θ(A))∂θ(A) < 0.

Thusθ(A) = 0.

(ii) For A ≥ Aimp, pH − γI +(ηp+(1− p))A− c− γA+μ(A−AL) > 0 and∂L(θ(A))∂θ(A) > 0.

Thusθ(A) = 1.

Aimp is defined by:∫∞Aimp

Af(A)dA − AL

∫∞Aimp

f(A)dA = 0. Let us defineG(A) =∫∞A Af(A)dA−AL

∫∞A f(A)dA. Note thatG(0) = E(A)−AL andG(AL) =

∫∞AL

Af(A)dA−

AL

∫∞AL

f(A)dA > 0. For this case to exist, we needE(A) < AL. Further,∂G(A)∂A = f(A)(AL −

A) > 0 and if E(A) < AL, Aimp is unique. To summarize, this case returns optimal measure-

ments whenE(A) < AL and prescribes firms aboveAL to measure while the remaining do not

measure.

CASE 3: μ = γ − (ηp + (1− p))

Lastly, when the constraint is binding andμ = γ − (ηp + (1− p)), we have:

(i) For A < AL, pH−γI +(ηp+(1−p))A− c−γA > 0 and∂L(θ(A))∂θ(A) > 0. Thusθ(A) = 1.

(ii) For AL < A < AH , ∂L(θ(A))∂θ(A) > 0. Thusθ(A) = 1.

(iii) For A ≥ AH , μ(AH −AL) > 0 and ∂L(θ(A))∂θ(A) > 0. Thusθ(A) = 1.

This case prescribes no measurement, which is never optimal and can be ruled out.

CASE 4: μ < γ − (ηp + (1− p))

When the constraint is binding andμ < γ − (ηp + (1− p)), there exists a uniqueAup > AH

such that

(i) For A < Aup, pH − γI + (ηp + (1− p))A− c− γA + μ(A−AL) > 0 and ∂L(θ(A))∂θ(A) > 0.

Thusθ(A) = 1.

(ii) For A ≥ Aup, pH − γI + (ηp + (1− p))A− c− γA + μ(A−AL) < 0 and ∂L(θ(A))∂θ(A) < 0.

Thusθ(A) = 0.

Aup is given by:∫ Aup

0 Af(A)dA−AL

∫ Aup

0 f(A)dA = 0. Let us defineM(A) =∫ A0 Af(A)dA−

AL

∫ A0 f(A)dA. As M(AL) =

∫ AL

0 Af(A)dA − AL

∫ AL

0 f(A)dA < 0 and whenA → ∞,

M(A) = E(A)−AL. To have this case we needE(A) > AL and∫ AH

0 Af(A)dA−AL

∫ AH

0 f(A)dA <

0. Under those conditions,Aup exists. FurtherM(A) is increasing as∂M(A)

∂A= f(A)(A−AL) ≥

0. ThusAup is unique. To summarize case 4 returns optimal measurements whenE(A) > AL

and∫ AH

0 Af(A)dA − AL

∫ AH

0 f(A)dA < 0, and prescribe firms to measure aboveAup and

the others not to measure. We turn to the comparative statics of the thresholds. The impair-

ment thresholdAimp is affected by a change in the exogenous parameters as follows:∂Aimp

∂q =

− ∂G(Aimp)/∂q∂G(Aimp)/∂Aimp

=(∂AL/∂q)

∫∞Aimp

f(A)da

f(Aimp)[AL−Aimp] , whereq = H, c, I, p, Δp, η, γ. Hence,∂Aimp

∂γ > 0,∂Aimp

∂I > 0, ∂Aimp

∂H < 0, ∂Aimp

∂c > 0, ∂Aimp

∂Δp < 0, ∂Aimp

∂η < 0, ∂Aimp

∂p < 0. We next show that

46

Aimp is increasing in a collateral squeeze. Let us define the functionGimp(X) =∫∞

X Af(A)dA∫∞X f(A)dA

.

The derivative is:G′imp(X) =

f(X)(∫∞

X Af(A) dA−X∫∞

X f(A) dA)(∫∞

X f(A) dA)2> 0. When there is a collateral

shock,Ashockimp is defined by:

∫∞Ashock

impAf(A)dA

∫∞Ashock

impf(A)dA

= AL1−ε > AL. ThusAshock

imp > Aimp asGimp(.) is

increasing.

The comparative statics is∂Aup

∂q = − ∂M(Aup)/∂q∂M(Aup)/∂Aup

= (∂AL/∂q)∫ Aup0 f(A)dA

f(A)[Aup−AL] , whereq =

H, c, I, p, Δp, η, γ. Hence,∂Aup

∂γ > 0, ∂Aup

∂I > 0, ∂Aup

∂H < 0, ∂Aup

∂c > 0, ∂Aup

∂Δp < 0, ∂Aup

∂η < 0,∂Aup

∂p < 0. We next show thatAup is increasing in a collateral squeeze. Let us define the func-

tion Glib(X) =∫X0 Af(A)dA∫X0 f(A)dA

. The derivative is:G′lib(X) =

f(X)(X∫X0 f(A) dA−

∫X0 Af(A) dA)

(∫X0 f(A) dA)2

> 0.

When there is a collateral shock,Ashockup is defined by:

∫Ashockup

0 Af(A)dA∫ Ashock

up0 f(A)dA

= AL1−ε > AL.. Thus

Ashockup > Aup asGlib(.) is increasing.

Proof of Corollary 1: Part (i): A write-up accounting system is optimal whenE(A) > AL,

which corresponds toγ < γL. We start by noting that:

• If Aup > AH then∫ AH

0 Af(A)dA−AL

∫ AH

0 f(A)dA < 0.

• If Aup < AH then∫ AH

0 Af(A)dA−AL

∫ AH

0 f(A)dA > 0.

• If Aup = AH then∫ AH

0 Af(A)dA−AL

∫ AH

0 f(A)dA = 0.

Next we note thatAup andAH are increasing and decreasing inγ, respectively and thatg(γ) ≡∫ AH(γ)0 Af(A)dA−AL(γ)

∫ AH(γ)0 f(A)dA is decreasing inγ, because∂g(γ)

∂γ = ∂AH(γ)∂γ f(AH(γ))[AH(γ)+

AL(γ)]− ∂AL(γ)∂γ

∫ AH(γ)0 f(A)dA < 0. Finally, we observe that:

• At γ → γmin, AH →∞ andg(γmin) =∫ AH

0 Af(A)dA−AL

∫ AH

0 f(A)dA = E(A)−AL.

If E(A) > AL,theng(γmin) > 0 and this turns out to be true when we have optimal write-

up measurements.

• At γ = γL, Aup → ∞ asAL(γL) = E(A) andAH(γL) = pH−γLI−cγL−(ηp+(1−p)) < Aup(γL). It

follows thatg(γL) =∫ AH(γL)0 Af(A)dA− E(A)

∫ AH(γL)0 f(A)dA < 0.

• At γ = γmax, AL(γmax) = AH(γmax) andg(γmax) =∫ AH(γmax)0 (A−AH(γmax))f(A)dA <

0.

Thus, there exists a uniqueγup ∈ (γmin,min(γL, γmax)), such thatg(γup) = 0, i.e. AH(γup) =

Aup(γup). When there is a collateral shock,g(γ) is redefined asgShock(γ) as followsgshock(AH(γ)) =∫ AH (γ)

1−ε

0 Af(A)dA− AL(γ)1−ε

∫ AH (γ)

1−ε

0 f(A)dA. Applying the Implicit function theorem,

47

∂γup

∂ε= −

∂gshock(γup)∂ε

∂gshock(γup)∂γup

= −AH(γup)f

(AH(γup)

1−ε

)(AH(γup)−AL(γup))− (1− ε)AL(γup)

∫ AH (γup)

1−ε

0 f(A)dA

(1− ε)

(∂AH(γup)

∂γupf(

AH(γup)(1−ε)

)(AH(γup)−AL(γup))−

∂AL(γup)∂γup

∫ AH (γup)

1−ε

0 f(A)dA

)

The denominator is negative, hence it follows that the comparative statics ofγup is determined bythe sign of the numerator. This, however, depends on the parameters in the economy. Specifically,

the numerator is equal to

AH(γup)f

(AH(γup)

1− ε

)

(AH(γup)−AL(γup))− (1− ε)AL(γup)∫ AH (γup)

1−ε

0f(A)dA

=(1− ε)AL(γup)

f(

AH(γup)(1−ε)

)

AH(γup)AL(γup)

.(AH(γup)−AL(γup))

(1− ε)−

∫ AH (γup)

1−ε

0 f(A)dA

f(

AH(γup)1−ε

)

.

Given that (1−ε)AL(γup)

f(

AH (γup)

(1−ε)

) > 0, the change inγup depends only on the sign of the expression

AH(γup)AL(γup) .

(AH(γup)−AL(γup))(1−ε) −

∫ AH (γup)1−ε

0 f(A)dA

f(

AH (γup)

1−ε

) .

Part (ii): If E(A) < AL, i.e. γ > γL the optimal information system is impairment-like.

Combining the conditions, requiring information system, we obtain result(ii) .

Part (iii): If γ > γmax all firms make the financing investment and hence, they are indifferent

between any information system.

Proof of Proposition 6: When setting the investment threshold the benevolent social planner

maximizes:

maxA

∫ A

0(pH + ηA− c)f(A)dA

s.t I

∫ A

0f(A)dA = K +

∫ ∞

AAf(A)dA

The solution to this program isA = A∗, whereA∗ satisfiesI∫ A∗

0 f(A)dA = K +∫∞A∗ Af(A)dA.

As long asK < I , thenA∗ <∞.

Proof of Corollary 2: Part (i): The thresholdA∗ is determined by the resource constraint.

Let denoteN(X) = I∫ X0 f(A)dA −

∫∞X Af(A)dA − K. Then applying the implicit function

theorem,∂A∗

∂I = −∂N(A∗)

∂I∂N(A∗)

∂A∗= −

∫ A∗

0 f(A)dA

f(A∗)(I+A∗) < 0. If there is a collateral squeeze decreasing the

assets’ value byε > 0, we defineNshock(X) = I∫ X0 f(A)dA − K −

∫∞X (1 − ε)Af(A)dA.

Then,∂A∗

shock∂ε = −

∂Nshock(A∗shock)

∂ε∂Nshock(A∗

shock)

∂A

= −∫ A∗

shock0 Af(A)dA

f(A∗shock)(I+A∗

shock(1−ε)) < 0. If there is a credit crunch

decreasingK by κ > 0, we defineNcrunch(X) = I∫ X0 f(A)dA − (1 − κ)K −

∫∞X Af(A)dA.

48

Then,∂A∗

crunch∂κ = −

∂Ncrunch(A∗crunch)

∂κ∂Ncrunch(A∗

crunch)

∂A

= − Kf(A∗

crunch)(I+A∗crunch(1−ε)) < 0.

Part (ii): It is straightforward to see that the total social surplus∫ A∗

0 (pH + ηA − c)f(A)dA

is increasing inp, H andη, and decreasing inc. SinceA∗ is increasing inK (decreasing inκ) and

decreasing inI it follows that the total social surplus is increasing inK and decreasing inI.

Proof of Proposition 7: The net demand without moral hazard is defined asNet(γ) =

I∫ AH(γ)0 f(A)dA − K −

∫∞AH(γ) Af(A)dA. and is decreasing inγ: ∂Net(γ)

∂γ = AH(γ)(1 +

I)∂AH(γ)∂γ f(AH(γ)) < 0. At γ → γmin, AH → ∞ andNet(γmin) = I − K > 0. Let define

γ0 = pH−cI , such thatAH(γ0) = 0. Note thatγ0 > γmax, becauseAL > 0 by assumption. Then,

atγ = γ0, Net(γmax) = −K − E(A) < 0. Hence, there exists a uniqueγFB ∈ (γmin, γ0), such

thatNet(γFB) = 0. By construction,AH(γFB) = A∗.

Proof of Corollary 3: The comparative statics ofγFB is as follows:∂γFB

∂q = −∂Net(AH (γFB))

∂q

∂Net(AH (γFB))

∂γ

=

−∂AH (γFB)

∂qf(AH(γFB))(I+AH(γFB))

∂AH (γFB)

∂γf(AH(γFB))(I+AH(γFB))

, whereq = H, c, p, η. Hence, ∂γFB

∂p > 0, ∂γFB

∂H > 0,

∂γFB

∂c < 0, ∂γFB

∂η > 0. For I, ∂γFB

∂I = −∫ AH (γFB)0 f(A)dA+

∂AH (γFB)

∂If(AH(γFB))(I+AH(γFB))

∂AH (γFB)

∂γf(AH(γFB))(I+AH(γFB))

. We

know that∂AH(γFB)∂I f(AH(γFB))(I+AH(γFB)) = − γ

γ−(ηp+(1−p))f(AH(γFB))(I+AH(γFB))

and hence∂γFB

∂I can be negative or positive. ForK, ∂γFB

∂K = 1∂AH (γFB)

∂γf(AH(γFB))(I+AH(γFB))

<

0. The cost of capitalγFB can offset the changes inp, H, η andc, so that the thresholdAH(γFB)

does not change,AH(γFB) = A∗ is not violated and the markets clear. As shown in Corollary 2

changes inI, aggregate wealth and outside capital affectA∗. The comparative statics ofAH(γ)

follow those ofA∗. AH(γFB) is decreasing inI, collateral squeeze and credit crunch.

Proof of Lemma 5: Part (i): We start by noting that forγ ∈ (γmin, γup) the net de-

mandQ(γ) = I∫ AH(γ)0 f(A)dA − K −

∫∞AH(γ) Af(A)dA is decreasing inγ, because∂Q(γ)

∂γ =

f(AH(γ))∂AH(γ)∂γ (I + AH(γ)) < 0. At γ = γmin, Q(γmin) = I −K > 0, becauseAH → ∞.

Next, we note that forγ ∈ (γup, min(γL, γmax)) the net demandT (γ) = I∫ Aup(γ)0 f(A)dA −

K −∫∞Aup(γ) Af(A)dA is increasing inγ, because∂T (γ)

∂γ = f(Aup(γ))∂Aup(γ)∂γ (I + Aup(γ)) > 0.

At γ = min(γL, γmax), T (min(γL, γmax)) = I − K > 0, becauseAup → ∞. At γ = γup,

Aup(γup) = AH(γup) and∫ AH(γup)0 Af(A)dA− AL(γup)

∫ AH(γup)0 Af(A)dA = 0. For a write-

up information system to be optimal it has to be the case thatQ(γup) = T (γup) < 0. Combining

the two conditions leads to the result(i).

Part (ii): We note that forγ ∈ (min(γL, γmax), γmax) the net demandZ(γ) = I∫∞Aimp(γ) f(A)dA−

K −∫ Aimp(γ)0 Af(A)dA is decreasing inγ: ∂Z(γ)

∂γ = −f(Aimp(γ))∂Aimp(γ)∂γ (I + Aimp(γ)) < 0.

If this net demand exists, atγ = γL, the net demandZ(γL) = T (γL) = I − K > 0. For an

impairment-like information system to be optimal it has to be the case thatZ(γmax) < 0. Combin-

ing this condition with the financing condition for non-disclosing firms,∫∞Aimp(γmax) Af(A)dA−

AL(γmax)∫∞Aimp(γmax) Af(A)dA = 0, leads to result(ii) .

49

Proof of Proposition 8:

• If E(A) > W low then γmin < γL. As shown forγ ∈ (γmin, γup) the net demand

Q(γ) is decreasing inγ and atγ = γmin, Q(γmin) = I − K > 0. As shown, ifI

AL(γup) <K+

∫∞AH (γup) Af(A)dA

∫Aup(γup)0 Af(A)dA

, thenQ(γup) < 0 and there exists a uniqueγ∗ < γup

such thatQ(γ∗) = 0. By construction,γ∗ = γFB .

• Also, as shown forγ ∈ (γup, min(γL, γmax)) the net demandT (γ) is increasing inγ and

at γ = min(γL, γmax), T (γ) = I − K > 0. If IAL(γup) <

K+∫∞

AH (γup) Af(A)dA∫ Aup(γup)0 Af(A)dA

, then

T (γup) < 0 and there exists a uniqueγ∗∗ ∈ (γmax, γup) such thatT (γ∗∗) = 0.

Hence, ifE(A) > W low and IAL(γup) <

K+∫∞

AH (γup) Af(A)dA∫ Aup(γup)0 Af(A)dA

, there existGE1 andGE2.

• If W low < E(A) ≤ W imp then γmin < γup < γL < γmax < γH . If IAL(γmax) ≥

K+∫ Aimp(γmax)

0 Af(A)dA∫∞Aimp(γmax) Af(A)dA

there exist a general equilibriumGE3, defined byZ(γ∗∗∗) = 0. To

see this start by recalling thatZ(γ) is decreasing inγ and thatZ(γL) = T (γL) = I −K >

0. Noting that if IAL(γmax) ≥

K+∫ Aimp(γmax)

0 Af(A)dA∫∞Aimp(γmax) Af(A)dA

, thenZ(γmax) < 0, we can conclude

that there exists a uniqueγ∗∗∗, such thatZ(γ∗∗∗) = 0.

• Finally, if E(A) ≤ W low, thenγL ≤ γmin < γmax < γH and only impairment-like

measurements can be optimal. As shown above,GE3, defined byZ(γ∗∗∗) = 0 exists

if IAL(γmax) ≥

K+∫ Aimp(γmax)

0 Af(A)dA∫∞Aimp(γmax) Af(A)dA

and IAL(γmin) ≤

K+∫ Aimp(γmin)

0 Af(A)dA∫∞Aimp(γmin) Af(A)dA

(so that

Z(γmin) > 0).

Proof of Corollary 4: The proof of Corollary 4 follows directly from the proof of Proposition

8 and is omitted.

Proof of Corollary 5: Comparative statics ofγ∗: γ∗ is defined byQ(γ∗) = 0, where

γmin < γ∗ < γup < min(max(γL, γmin), γmax). Applying the Implicit function Theorem,

∂γ∗

∂q = −∂Q(γ∗)

∂q∂Q(γ∗)

∂γ

= −∂AH (γ∗)

∂qf(AH(γ∗))(I+AH(γ∗))

∂AH (γ∗)

∂γf(AH(γ∗))(I+AH(γ∗))

whereq = H, c, p, η. Hence,∂γ∗

∂p > 0, ∂γ∗

∂H >

0, ∂γ∗

∂c < 0, ∂γ∗

∂η > 0.For I, ∂γ∗

∂I = −∂Q(γ∗)

∂I∂Q(γ∗)

∂γ

= −∫ AH (γ∗)0 f(A)dA+

∂AH (γ∗)

∂If(AH(γ∗))(I+AH(γ∗))

∂AH (γ∗)

∂γf(AH(γ∗))(I+AH(γ∗))

.

We know that: ∂AH(γ∗)∂I f(AH(γ∗))(I + AH(γ∗)) = − γ

γ−(ηp+(1−p))f(AH(γ∗))(I + AH(γ∗))

and hence the sign of∂γ∗

∂I is ambiguous. By constructionAH(γ∗) = A∗ and thus has the same

comparative statics asA∗.

Comparative statics ofγ∗∗: γ∗∗ is defined byT (γ∗∗) = 0, whereγmin < γup < γ∗∗ <

min(max(γL, γmin), γmax). Let denote

50

Ψ1(Aup, γ) =∫ Aup

0Af(A)dA−AL(γ)

∫ Aup

0f(A)dA

Ψ2(Aup, γ) = I

∫ Aup

0f(A)dA−K −

∫ ∞

Aup

Af(A)dA

We define the JacobianJlib as follows:Jlib =

[(Aup −AL(γ))f(Aup) −

∂AL(γ)∂γ

∫ Aup

0 f(A)dA

(Aup + I)f(Aup) 0

]

.

Applying the Implicit Function Theorem yields:

[∂Aup

∂q∂γ∗∗

∂q

]

= −J−1lib ×

[∂Ψ1(Aup,γ)

∂q∂Ψ2(Aup,γ)

∂q

]

, whereq

is equal to any exogenous parameter in the economy.

If q is p, H, η or c, ∂Ψ2(Aup,γ)∂q = 0. If q = I, ∂Ψ2(Aup,γ)

∂I =∫ Aup

0 f(A)dA. Simplifying, if

q = p or H or η or c:

[∂Aup

∂q∂γ∗∗

∂q

]

=

0

−∂AL∂q

∂AL(γ)

∂γ

and ∂γ∗∗

∂q follows the sign of−∂AL∂q . If q = I:

[∂Aup

∂q∂γ∗∗

∂q

]

=

−∫ Aup0 f(A)dA

(Aup+I)f(Aup) < 0

−Aup−AL+

∂AL∂I

(Aup+I)∂AL(γ)

∂γ(Aup+I)

< 0

.

Comparative statics ofγ∗∗∗: γ∗∗∗ ∈ (min(max(γL, γmin), γmax), γmax) is defined byZ(γ∗∗∗) =

0. Let denote

Ψ3(Aimp, γ) =∫ ∞

Aimp

Af(A)dA−AL(γ)∫ ∞

Aimp

f(A)dA

Ψ4(Aimp, γ) = I

∫ ∞

Aimp

f(A)dA−K −∫ Aimp

0Af(A)dA

We define the JacobianJimp as followsJimp =

[(AL(γ)−Aimp)f(Aimp) −

∂AL(γ)∂γ

∫∞Aimp

f(A)dA

−(Aimp + I)f(Aimp) 0

]

.

Applying the Implicit Function Theorem yields

[∂Aimp

∂q∂γ∗∗∗

∂q

]

= −J−1imp×

[∂Ψ3(Aimp,γ)

∂q∂Ψ4(Aimp,γ)

∂q

]

, whereq

is equal to any exogenous parameter in the economy.Ifq is p, H, η or c, ∂Ψ4(Aimp,γ)∂q = 0. If q = I,

∂Ψ4(Aimp,γ)∂I =

∫∞Aimp

f(A)dA. Simplifying, if q = p or H or η or c,

[∂Aimp

∂q∂γ∗∗∗

∂q

]

=

0

−∂AL∂q

∂AL(γ)

∂γ

and∂γ∗∗∗

∂q follows the sign of−∂AL∂q . Forq = I,

[∂Aimp

∂q∂γ∗∗∗

∂q

]

=

∫∞Aimp

f(A)dA

(Aimp+I)f(Aimp) > 0

−Aimp−AL+(Aimp+I)

∂AL∂I

∂AL(γ)

∂γ(Aimp+I)

< 0

,

because−(Aimp − AL + (Aimp + I)∂AL∂I ) = −( γ

ηp+(1−p) + 1)Aimp + AL −γ

ηp+(1−p)I =

−( γηp+(1−p) + 1)Aimp − 1

ηp+(1−p)(pH − p cΔp) < 0.

Proof of Corollary 6: We use the Walrasian tatonnement process to investigate the stability

of the general equilibria. Let us assume that there is a small perturbationδ > 0.

51

GE1: – The small perturbation is leading to a higher (lower) cost of capitalγ + δ > γ∗ (resp.

γ− δ < γ∗). At γ + δ (resp.γ− δ), there is excess supply (excess demand) and to fix

it, as the net demandQ is decreasing, the cost of capital needs to be lower (higher).

⇒ GE1 is a stable equilibrium as in response to a small perturbation on the cost of

capital, the net demand adjusts so that we converge back to the equilibrium cost of

capitalγ∗.

GE2: – The small perturbation is leading to a higher (lower) cost of capitalγ +δ > γ∗∗ (resp.

γ − δ < γ∗∗). At γ + δ (resp.γ − δ), there is excess supply (excess demand) and to

fix it, as the net demandT is increasing, the cost of capital needs to be higher (lower).

⇒ GE2 is not a stable equilibrium as in response to a small perturbation on the cost of

capital, the net demand adjusts so that we diverge from the equilibrium cost of capital

γ∗∗.

GE3: – The small perturbation is leading to a higher (lower) cost of capitalγ + δ > γ∗∗∗

(resp.γ − δ < γ∗∗∗). At γ + δ (resp.γ − δ), there is excess supply (excess demand)

and to fix it, as the net demandZ is decreasing, the cost of capital needs to be lower

(higher).

⇒ GE1 is a stable equilibrium as in response to a small perturbation on the cost of

capital, the net demand adjusts so that we converge back to the equilibrium cost of

capitalγ∗∗∗.

Proof of Corollary 7: When there is a collateral squeeze,

Qshock(γ) = I

∫ AH(γ)/(1−ε)

0f(A)dA−K −

∫ ∞

AH(γ)/(1−ε)(1− ε)Af(A)dA,

T shock(γ) = I

∫ Aup(γ)

0f(A)dA−K −

∫ ∞

Aup(γ)(1− ε)Af(A)dA,

Zshock(γ) = I

∫ ∞

Aimp(γ)f(A)dA−K −

∫ Aimp(γ)

0(1− ε)Af(A)dA.

Ψshock1 (Aup, γ) =

∫ Aup

0(1− ε)Af(A)dA−AL(γ)

∫ Aup

0f(A)dA

Ψshock2 (Aup, γ) = I

∫ Aup

0f(A)dA−K −

∫ ∞

Aup

(1− ε)Af(A)dA

Ψshock3 (Aimp, γ) =

∫ ∞

Aimp

(1− ε)Af(A)dA−AL(γ)∫ ∞

Aimp

f(A)dA

Ψshock4 (Aimp, γ) = I

∫ ∞

Aimp

f(A)dA−K −∫ Aimp

0(1− ε)Af(A)dA

52

Now Jshocklib =

[(Aup(1− ε)−AL(γ))f(Aup) −

∂AL(γ)∂γ

∫ Aup

0 f(A)dA

(Aup(1− ε) + I)f(Aup) 0

]

and

Jshockimp =

[(AL(γ)−Aimp(1− ε))f(Aimp) −

∂AL(γ)∂γ

∫∞Aimp

f(A)dA

−(Aimp(1− ε) + I)f(Aimp) 0

]

.

The comparative statics ofγ∗ is ∂γ∗

∂ε = −∂Qshock(γ)

∂ε∂Qshock(γ)

∂γ

= −AHf(A/(1−ε))

(1−ε)2(I+AH(1−ε))+

∫∞AH (γ∗) Af(A)dA

∂AH∂γ

(I+AH(γ∗))f(AH(γ∗))>

0, andAH(γ∗) decreases.

For γ∗∗ andAup(γ∗∗) the comparative statics is

[∂Aup

∂ε∂γ∗∗

∂ε

]

= −Jshock −1lib ×

[∂Ψshock

1 (Aup,γ)∂ε

∂Ψshock2 (Aup,γ)

∂ε

]

.

Simplifying,

[∂Aup

∂ε∂γ∗∗

∂ε

]

=

−∫∞

AupAf(A)dA

f(A)(Aup(1−ε)+I) < 0

−(Aup(1−ε)+I)

∫ Aup0 Af(A)dA+(Aup(1−ε)−AL)

∫∞Aup

Af(A)dA

∂AL∂γ

(Aup(1−ε)+I)∫Aup0 f(A)dA

< 0

. For

γ∗∗∗ andAup(γ∗∗∗) the comparative statics is

[∂Aimp

∂ε∂γ∗∗∗

∂ε

]

= −Jshock −1imp ×

[∂Ψshock

3 (Aimp,γ)∂ε

∂Ψshock4 (Aimp,γ)

∂ε

]

.

Simplifying,

[∂Aimp

∂ε∂γ∗∗∗

∂ε

]

=

∫Aimp0 Af(A)dA

f(A)(Aimp(1−ε)+I) > 0

−(AL−Aimp(1−ε))

∫Aimp0 Af(A)dA−(Aimp(1−ε)+I)

∫∞Aimp

Af(A)dA

∂AL∂γ

(Aimp(1−ε)+I)∫∞

Aimpf(A)dA

> 0

,

because

M(γ∗∗∗) = (AL −Aimp(1− ε))∫ Aimp

0Af(A)dA− (Aimp(1− ε) + I)

∫ ∞

Aimp

Af(A)dA

= AL

(∫ Aimp

0Af(A)dA− I

∫ ∞

Aimpf(A)dA

)

−AimpE(A)

= −KAL −AimpE(A)

< 0

When there is a credit crunch,

Qcrunch(γ) = I

∫ AH(γ)

0f(A)dA− (1− κ)K −

∫ ∞

AH(γ)Af(A)dA,

T crunch(γ) = I

∫ Aup(γ)

0f(A)dA− (1− κ)K −

∫ ∞

Aup(γ)Af(A)dA,

Zcrunch(γ) = I

∫ ∞

Aimp(γ)f(A)dA− (1− κ)K −

∫ Aimp(γ)

0Af(A)dA

Ψcrunch2 (Aup, γ) = I

∫ Aup

0f(A)dA− (1− κ)K −

∫ ∞

Aup

Af(A)dA

Ψcrunch4 (Aimp, γ) = I

∫ ∞

Aimp

f(A)dA− (1− κ)K −∫ Aimp

0Af(A)dA

For γ∗ the comparative statics is∂γ∗

∂κ = −∂Qcrunch(γ)

∂κ∂Qcrunch(γ)

∂γ

= − 1∂AH

∂γ(I+AH(γ∗))f(AH(γ∗))

> 0, and

53

AH(γ∗) decreases. Forγ∗∗ and Aup(γ∗∗) the comparative statics is

[∂Aup

∂κ∂γ∗∗

∂κ

]

= −J−1lib ×

[∂Ψ1(Aup,γ)

∂κ∂Ψcrunch

2 (Aup,γ)∂κ

]

, whereJ−1lib is as defined in the proof to Corollary 5. This is equivalent

to

[∂Aup

∂κ∂γ∗∗

∂κ

]

=

−{(Aup + I)f(Aup)}−1 < 0

− Aup−AL∂AL∂γ

(Aup+I)∫ Aup0 f(A)dA

< 0

.

Forγ∗∗∗ andAup(γ∗∗∗) the comparative statics is

[∂Aimp

∂κ∂γ∗∗∗

∂κ

]

= −J−1imp ×

[∂Ψ3(Aimp,γ)

∂κ∂Ψcrunch

4 (Aimp,γ)∂κ

]

,

whereJ−1imp is as defined in the proof to Corollary 5.

This is equivalent to

[∂Aimp

∂κ∂γ∗∗∗

∂κ

]

=

{(Aimp + I)f(Aimp)}−1 > 0

− Aimp−AL∂AL∂γ

(Aimp+I)∫∞

Aimpf(A)dA

> 0

.

54

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